diff --git a/test/book/cornerstones/algebraic-geometry/alexander-surface.jlcon b/test/book/cornerstones/algebraic-geometry/alexander-surface.jlcon
index e11ef69384e1..3639a00b22d9 100644
--- a/test/book/cornerstones/algebraic-geometry/alexander-surface.jlcon
+++ b/test/book/cornerstones/algebraic-geometry/alexander-surface.jlcon
@@ -7,7 +7,7 @@ julia> degree(X)
 9
 
 julia> S = ambient_coordinate_ring(X)
-Multivariate polynomial ring in 5 variables over GF(31991) graded by
+Multivariate polynomial ring in 5 variables over GF(31991) graded by 
   x -> [1]
   y -> [1]
   z -> [1]
diff --git a/test/book/cornerstones/algebraic-geometry/circlepar.jlcon b/test/book/cornerstones/algebraic-geometry/circlepar.jlcon
index 3e92366f3372..d7a017457722 100644
--- a/test/book/cornerstones/algebraic-geometry/circlepar.jlcon
+++ b/test/book/cornerstones/algebraic-geometry/circlepar.jlcon
@@ -4,22 +4,22 @@ julia> I = ideal(R, [x^2 + y^2 - 1, y - t*x - 1]);
 
 julia> Gy = groebner_basis(I, ordering = lex([y, x, t]), complete_reduction=true)
 Gröbner basis with elements
-  1 -> x^2*t^2 + x^2 + 2*x*t
-  2 -> y - x*t - 1
+1 -> x^2*t^2 + x^2 + 2*x*t
+2 -> y - x*t - 1
 with respect to the ordering
-  lex([y, x, t])
+lex([y, x, t])
 
 julia> factor(Gy[1])
 1 * (x*t^2 + x + 2*t) * x
 
 julia> Gx = groebner_basis(I, ordering = lex([x, y, t]), complete_reduction=true)
 Gröbner basis with elements
-  1 -> y^2*t^2 + y^2 - 2*y - t^2 + 1
-  2 -> x*t - y + 1
-  3 -> x*y - x + y^2*t - t
-  4 -> x^2 + y^2 - 1
+1 -> y^2*t^2 + y^2 - 2*y - t^2 + 1
+2 -> x*t - y + 1
+3 -> x*y - x + y^2*t - t
+4 -> x^2 + y^2 - 1
 with respect to the ordering
-  lex([x, y, t])
+lex([x, y, t])
 
 julia> factor(Gx[1])
 1 * (y - 1) * (y*t^2 + y + t^2 - 1)
diff --git a/test/book/cornerstones/algebraic-geometry/ex11.jlcon b/test/book/cornerstones/algebraic-geometry/ex11.jlcon
index 3f903771e98b..1fdb5883036c 100644
--- a/test/book/cornerstones/algebraic-geometry/ex11.jlcon
+++ b/test/book/cornerstones/algebraic-geometry/ex11.jlcon
@@ -4,8 +4,8 @@ julia> I = ideal(R, [x^2+y^2+2*z^2-8, x^2-y^2-z^2+1, x-y+z]);
 
 julia> groebner_basis(I, ordering = lex(R), complete_reduction = true)
 Gröbner basis with elements
-  1 -> 6*z^4 - 18*z^2 + 1
-  2 -> y + 3*z^3 - 9*z
-  3 -> x + 3*z^3 - 8*z
+1 -> 6*z^4 - 18*z^2 + 1
+2 -> y + 3*z^3 - 9*z
+3 -> x + 3*z^3 - 8*z
 with respect to the ordering
-  lex([x, y, z])
+lex([x, y, z])
diff --git a/test/book/cornerstones/algebraic-geometry/ex21a.jlcon b/test/book/cornerstones/algebraic-geometry/ex21a.jlcon
index 4bf42c540163..01a275352dd2 100644
--- a/test/book/cornerstones/algebraic-geometry/ex21a.jlcon
+++ b/test/book/cornerstones/algebraic-geometry/ex21a.jlcon
@@ -1,4 +1,4 @@
-julia> J1 = ideal([H(I[1]), H(I[2])])
+julia> J1 = ideal([H(I[1]), H(I[2])]) 
 Ideal generated by
   x0*x2 - x1^2
   x0*x3 - x1*x2
diff --git a/test/book/cornerstones/algebraic-geometry/ex23a.jlcon b/test/book/cornerstones/algebraic-geometry/ex23a.jlcon
index 67637c6f9acf..eaca161def65 100644
--- a/test/book/cornerstones/algebraic-geometry/ex23a.jlcon
+++ b/test/book/cornerstones/algebraic-geometry/ex23a.jlcon
@@ -12,11 +12,10 @@ julia> other_pos_abs = pos_abs == 1 ? 2 : 1
 julia> cI2 = cIabs[other_pos_abs][3]
 Ideal generated by
   648*y + (-160*_a^3 - 1269*_a^2 + 22446*_a + 972)*z
-  184147758075888*x + (2850969000960*_a^3 + 22611747888864*_a^2 - 399955313722176*_a - 17319636680832)*y + (2884707374400*_a^3 + 2
-  2782606070410*_a^2 - 405172045313820*_a - 57843867366864)*z
+  184147758075888*x + (2850969000960*_a^3 + 22611747888864*_a^2 - 399955313722176*_a - 17319636680832)*y + (2884707374400*_a^3 + 22782606070410*_a^2 - 405172045313820*_a - 57843867366864)*z
 
 julia> R2 = base_ring(cI2)
-Multivariate polynomial ring in 3 variables over number field graded by
+Multivariate polynomial ring in 3 variables over number field graded by 
   x -> [1]
   y -> [1]
   z -> [1]
diff --git a/test/book/cornerstones/algebraic-geometry/ex314.jlcon b/test/book/cornerstones/algebraic-geometry/ex314.jlcon
index 3a15caec9592..d387821bc9bc 100644
--- a/test/book/cornerstones/algebraic-geometry/ex314.jlcon
+++ b/test/book/cornerstones/algebraic-geometry/ex314.jlcon
@@ -68,13 +68,11 @@ M1 -> M
 e[1] -> (-x_0*x_3 + x_1*x_2)*e[1]
 Graded module homomorphism of degree [2]
 
-
 julia> phi2 = psi(tohomM1M(hom1[2]))
 M1 -> M
 e[1] -> (-x_0*x_2 + x_1^2)*e[1]
 Graded module homomorphism of degree [2]
 
-
 julia> kerphi2, _ = kernel(phi2);
 
 julia> iszero(kerphi2)
diff --git a/test/book/cornerstones/algebraic-geometry/exres.jlcon b/test/book/cornerstones/algebraic-geometry/exres.jlcon
index 087305294474..b6d52ae6d476 100644
--- a/test/book/cornerstones/algebraic-geometry/exres.jlcon
+++ b/test/book/cornerstones/algebraic-geometry/exres.jlcon
@@ -1,25 +1,25 @@
-julia> R, (w, x, y, z) = graded_polynomial_ring(QQ, ["w", "x", "y", "z"]);
+julia> S, (w, x, y, z) = graded_polynomial_ring(QQ, ["w", "x", "y", "z"]);
 
-julia> I = ideal([w^2-x*z,w*x-y*z,x^2-w*y,x*y-z^2,y^2-w*z]);
+julia> J = ideal([w^2-x*z,w*x-y*z,x^2-w*y,x*y-z^2,y^2-w*z]);
 
-julia> A, _ = quo(R, I);
+julia> A, _ = quo(S, J);
 
 julia> FA = free_resolution(A)
 Free resolution of A
-R^1 <---- R^5 <---- R^6 <---- R^2 <---- 0
+S^1 <---- S^5 <---- S^6 <---- S^2 <---- 0
 0         1         2         3         4
 
 julia> FA[1]
-Graded free module R^5([-2]) of rank 5 over R
+Graded free module S^5([-2]) of rank 5 over S
 
 julia> FA[2]
-Graded free module R^5([-3]) + R^1([-4]) of rank 6 over R
+Graded free module S^5([-3]) + S^1([-4]) of rank 6 over S
 
 julia> FA[3]
-Graded free module R^1([-4]) + R^1([-5]) of rank 2 over R
+Graded free module S^1([-4]) + S^1([-5]) of rank 2 over S
 
 julia> map(FA,1)
-R^5 -> R^1
+S^5 -> S^1
 e[1] -> (-w*z + y^2)*e[1]
 e[2] -> (x*y - z^2)*e[1]
 e[3] -> (-w*y + x^2)*e[1]
@@ -28,7 +28,7 @@ e[5] -> (w^2 - x*z)*e[1]
 Homogeneous module homomorphism
 
 julia> map(FA,2)
-R^6 -> R^5
+S^6 -> S^5
 e[1] -> -x*e[1] + y*e[2] - z*e[4]
 e[2] -> w*e[1] - x*e[2] + y*e[3] + z*e[5]
 e[3] -> -w*e[3] + x*e[4] - y*e[5]
@@ -38,7 +38,7 @@ e[6] -> (-w^2 + x*z)*e[1] + (-w*z + y^2)*e[5]
 Homogeneous module homomorphism
 
 julia> map(FA,3)
-R^2 -> R^6
+S^2 -> S^6
 e[1] -> -w*e[2] - y*e[3] + x*e[4] - e[6]
 e[2] -> (-w^2 + x*z)*e[1] + y*z*e[2] + z^2*e[3] - w*y*e[4] + (w*z - y^2)*e[5] + x*e[6]
 Homogeneous module homomorphism
diff --git a/test/book/cornerstones/algebraic-geometry/hilbert-polynomial.jlcon b/test/book/cornerstones/algebraic-geometry/hilbert-polynomial.jlcon
index f35655f922e3..3bba333557af 100644
--- a/test/book/cornerstones/algebraic-geometry/hilbert-polynomial.jlcon
+++ b/test/book/cornerstones/algebraic-geometry/hilbert-polynomial.jlcon
@@ -1,6 +1,6 @@
-julia> R, (w,x,y,z) = graded_polynomial_ring(QQ, ["w", "x", "y", "z"]);
+julia> S, (w,x,y,z) = graded_polynomial_ring(QQ, ["w", "x", "y", "z"]);
 
-julia> I = ideal(R, [x*w-y*z, y*w-(x-z)*(x-2*z)]);
+julia> I = ideal(S, [x*w-y*z, y*w-(x-z)*(x-2*z)]);
 
 julia> Q = projective_scheme(I);
 
diff --git a/test/book/cornerstones/algebraic-geometry/param.jlcon b/test/book/cornerstones/algebraic-geometry/param.jlcon
index b8e9dbe6e490..1e975acf1854 100644
--- a/test/book/cornerstones/algebraic-geometry/param.jlcon
+++ b/test/book/cornerstones/algebraic-geometry/param.jlcon
@@ -4,8 +4,7 @@ julia> f = x^5 + 10*x^4*y + 20*x^3*y^2 + 130*x^2*y^3 - 20*x*y^4 + 20*y^5 - 2*x^4
 
 julia> C = plane_curve(f)
 Projective plane curve
-  defined by 0 = x^5 + 10*x^4*y - 2*x^4*z + 20*x^3*y^2 - 40*x^3*y*z + x^3*z^2 + 130*x^2*y^3 - 150*x^2*y^2*z + 30*x^2*y*z^2 - 20*x*
-  y^4 - 90*x*y^3*z + 110*x*y^2*z^2 + 20*y^5 - 40*y^4*z + 20*y^3*z^2
+  defined by 0 = x^5 + 10*x^4*y - 2*x^4*z + 20*x^3*y^2 - 40*x^3*y*z + x^3*z^2 + 130*x^2*y^3 - 150*x^2*y^2*z + 30*x^2*y*z^2 - 20*x*y^4 - 90*x*y^3*z + 110*x*y^2*z^2 + 20*y^5 - 40*y^4*z + 20*y^3*z^2
 
 julia> conics = [x^2-x*z, y^2-y*z];
 
diff --git a/test/book/cornerstones/groups/actions.jlcon b/test/book/cornerstones/groups/actions.jlcon
index af50427d42b8..1083ae5aa2b9 100644
--- a/test/book/cornerstones/groups/actions.jlcon
+++ b/test/book/cornerstones/groups/actions.jlcon
@@ -3,15 +3,14 @@ G-set of
   Alt(4)
   with seeds 1:4
 
-julia> G = dihedral_group(6)
-Pc group of order 6
+julia> G = dihedral_group(6);
 
 julia> U = sub(G, [g for g in gens(G) if order(g) == 2])[1]
-Sub-pc group of order 2
+Pc group of order 2
 
 julia> r = right_cosets(G, U)
 Right cosets of
-  sub-pc group of order 2 in
+  pc group of order 2 in
   pc group of order 6
 
 julia> acting_group(r)
@@ -37,7 +36,7 @@ Group homomorphism
 julia> phi(G[1]), phi(G[2])
 ((2,3), (1,2,3))
 
-julia> function optimal_perm_rep(G)
+julia> function optimal_transitive_perm_rep(G)
          is_natural_symmetric_group(G) && return hom(G,G,gens(G))
          is_natural_alternating_group(G) && return hom(G,G,gens(G))
          cand = []  # pairs (U,h) with U ≤ G and h a map G -> Sym(G/U)
@@ -52,7 +51,7 @@ julia> function optimal_perm_rep(G)
 julia> U = dihedral_group(8)
 Pc group of order 8
 
-julia> optimal_perm_rep(U)
+julia> optimal_transitive_perm_rep(U)
 Group homomorphism
   from pc group of order 8
   to permutation group of degree 4 and order 8
@@ -60,13 +59,13 @@ Group homomorphism
 julia> isomorphism(PermGroup, U)
 Group homomorphism
   from pc group of order 8
-  to permutation group of degree 4 and order 8
+  to permutation group of degree 8 and order 8
 
 julia> permutation_group(U)
-Permutation group of degree 4 and order 8
+Permutation group of degree 8 and order 8
 
-julia> for g in all_transitive_groups(degree => 3:9, !is_primitive)
-         h = image(optimal_perm_rep(g))[1]
+julia> for g in all_transitive_groups(degree => 3:8, !is_primitive)
+         h = image(optimal_transitive_perm_rep(g))[1]
          if degree(h) < degree(g)
            id = transitive_group_identification(g)
            id_new = transitive_group_identification(h)
@@ -81,5 +80,3 @@ julia> for g in all_transitive_groups(degree => 3:9, !is_primitive)
 (8, 13) => (6, 6)
 (8, 14) => (4, 5)
 (8, 24) => (6, 11)
-(9, 4) => (6, 5)
-(9, 8) => (6, 9)
diff --git a/test/book/cornerstones/groups/chars.jlcon b/test/book/cornerstones/groups/chars.jlcon
index 0dc41d966ea9..a1c43f0089d9 100644
--- a/test/book/cornerstones/groups/chars.jlcon
+++ b/test/book/cornerstones/groups/chars.jlcon
@@ -24,7 +24,7 @@ julia> all(i -> s[i] == 0, ord5)
 true
 
 julia> (x -> [x["1a"], x["2b"], x["3a"], x["3b"]]).(filt)
-2-element Vector{Vector{QQAbFieldElem{AbsSimpleNumFieldElem}}}:
+2-element Vector{Vector{QQAbElem{AbsSimpleNumFieldElem}}}:
  [100, 0, 10, 4]
  [1800, 20, 0, 6]
 
diff --git a/test/book/cornerstones/groups/explSL25.jlcon b/test/book/cornerstones/groups/explSL25.jlcon
index ed265be8f3b3..62ecf567ff0b 100644
--- a/test/book/cornerstones/groups/explSL25.jlcon
+++ b/test/book/cornerstones/groups/explSL25.jlcon
@@ -4,21 +4,37 @@ SL(2,5)
 julia> T = character_table(G)
 Character table of SL(2,5)
 
-  2  3                 1                 1  3                  1                  1  1  1  2
-  3  1                 .                 .  1                  .                  .  1  1  .
-  5  1                 1                 1  1                  1                  1  .  .  .
- 
-    1a               10a               10b 2a                 5a                 5b 3a 6a 4a
- 
-X_1  1                 1                 1  1                  1                  1  1  1  1
-X_2  2 z_5^3 + z_5^2 + 1    -z_5^3 - z_5^2 -2 -z_5^3 - z_5^2 - 1      z_5^3 + z_5^2 -1  1  .
-X_3  2    -z_5^3 - z_5^2 z_5^3 + z_5^2 + 1 -2      z_5^3 + z_5^2 -z_5^3 - z_5^2 - 1 -1  1  .
-X_4  3    -z_5^3 - z_5^2 z_5^3 + z_5^2 + 1  3     -z_5^3 - z_5^2  z_5^3 + z_5^2 + 1  .  . -1
-X_5  3 z_5^3 + z_5^2 + 1    -z_5^3 - z_5^2  3  z_5^3 + z_5^2 + 1     -z_5^3 - z_5^2  .  . -1
-X_6  4                -1                -1  4                 -1                 -1  1  1  .
-X_7  4                 1                 1 -4                 -1                 -1  1 -1  .
-X_8  5                 .                 .  5                  .                  . -1 -1  1
-X_9  6                -1                -1 -6                  1                  1  .  .  .
+  2  3                 1                 1  3                  1
+  3  1                 .                 .  1                  .
+  5  1                 1                 1  1                  1
+
+    1a               10a               10b 2a                 5a
+
+X_1  1                 1                 1  1                  1
+X_2  2 z_5^3 + z_5^2 + 1    -z_5^3 - z_5^2 -2 -z_5^3 - z_5^2 - 1
+X_3  2    -z_5^3 - z_5^2 z_5^3 + z_5^2 + 1 -2      z_5^3 + z_5^2
+X_4  3    -z_5^3 - z_5^2 z_5^3 + z_5^2 + 1  3     -z_5^3 - z_5^2
+X_5  3 z_5^3 + z_5^2 + 1    -z_5^3 - z_5^2  3  z_5^3 + z_5^2 + 1
+X_6  4                -1                -1  4                 -1
+X_7  4                 1                 1 -4                 -1
+X_8  5                 .                 .  5                  .
+X_9  6                -1                -1 -6                  1
+
+  2                  1  1  1  2
+  3                  .  1  1  .
+  5                  1  .  .  .
+
+                    5b 3a 6a 4a
+
+X_1                  1  1  1  1
+X_2      z_5^3 + z_5^2 -1  1  .
+X_3 -z_5^3 - z_5^2 - 1 -1  1  .
+X_4  z_5^3 + z_5^2 + 1  .  . -1
+X_5     -z_5^3 - z_5^2  .  . -1
+X_6                 -1  1  1  .
+X_7                 -1  1 -1  .
+X_8                  . -1 -1  1
+X_9                  1  .  .  .
 
 julia> R = gmodule(T[end])
 G-module for G acting on vector space of dimension 6 over abelian closure of Q
@@ -30,7 +46,7 @@ julia> schur_index(T[end])
 2
 
 julia> gmodule_minimal_field(S)
-G-module for G acting on vector space of dimension 6 over number field
+G-module for G acting on vector space of dimension 6 over number field of degree 4 over QQ
 
 julia> B, mB = relative_brauer_group(base_ring(S), character_field(S));
 
diff --git a/test/book/cornerstones/groups/intro.jlcon b/test/book/cornerstones/groups/intro.jlcon
index 20e0e4f98c9a..e2d3574fee67 100644
--- a/test/book/cornerstones/groups/intro.jlcon
+++ b/test/book/cornerstones/groups/intro.jlcon
@@ -62,18 +62,12 @@ julia> pts = collect(orb)
 
 julia> visualize(convex_hull(pts))
 
-julia> R2 = free_module(K, 2) # the "euclidean" plane over K
-Vector space of dimension 2 over QQBar
-
-julia> A = R2([0,1])
-(Root 0 of x, Root 1.00000 of x - 1)
-
-julia> pts = [A*mat_rot^i for i in 0:4];
+julia> R2 = vector_space(K, 2); # the "euclidean" plane over K
 
 julia> sigma_1 = hom(R2, R2, [-R2[1], R2[2]])
 Module homomorphism
-  from vector space of dimension 2 over QQBar
-  to vector space of dimension 2 over QQBar
+  from vector space of dimension 2 over field of algebraic numbers
+  to vector space of dimension 2 over field of algebraic numbers
 
 julia> rot = hom(R2, R2, mat_rot);
 
@@ -127,10 +121,10 @@ julia> describe(H)
 "a finitely presented group"
 
 julia> H1, _ = derived_subgroup(H)
-(Sub-finitely presented group, Hom: H1 -> H)
+(Finitely presented group, Hom: H1 -> H)
 
 julia> H2, _ = derived_subgroup(H1)
-(Sub-finitely presented group, Hom: H2 -> H1)
+(Finitely presented group, Hom: H2 -> H1)
 
 julia> describe(H2)
 "Z x Z"
diff --git a/test/book/cornerstones/groups/reps.jlcon b/test/book/cornerstones/groups/reps.jlcon
index a2651f37f45e..792bd17669d9 100644
--- a/test/book/cornerstones/groups/reps.jlcon
+++ b/test/book/cornerstones/groups/reps.jlcon
@@ -12,9 +12,9 @@ julia> M = regular_gmodule(GF(7), G);
 
 julia> C = composition_factors_with_multiplicity(M)
 3-element Vector{Any}:
- (G-module for G acting on vector space of dimension 1 over GF(7), 1)
- (G-module for G acting on vector space of dimension 1 over GF(7), 1)
- (G-module for G acting on vector space of dimension 4 over GF(7), 2)
+ (G-module for G acting on vector space of dimension 1 over prime field of characteristic 7, 1)
+ (G-module for G acting on vector space of dimension 1 over prime field of characteristic 7, 1)
+ (G-module for G acting on vector space of dimension 4 over prime field of characteristic 7, 2)
 
 julia> [is_absolutely_irreducible(x[1]) for x in C]
 3-element Vector{Bool}:
@@ -28,12 +28,12 @@ Morphism of finite fields
   to finite field of degree 2 and characteristic 7
 
 julia> M = extension_of_scalars(C[3][1], phi)
-G-module for G acting on vector space of dimension 4 over GF(7, 2)
+G-module for G acting on vector space of dimension 4 over finite field of degree 2 and characteristic 7
 
 julia> composition_factors_with_multiplicity(M)
 2-element Vector{Any}:
- (G-module for G acting on vector space of dimension 2 over GF(7, 2), 1)
- (G-module for G acting on vector space of dimension 2 over GF(7, 2), 1)
+ (G-module for G acting on vector space of dimension 2 over finite field of degree 2 and characteristic 7, 1)
+ (G-module for G acting on vector space of dimension 2 over finite field of degree 2 and characteristic 7, 1)
 
 julia> G = pc_group(symmetric_group(4));
 
@@ -63,7 +63,7 @@ julia> pushfirst!(im, hom(FF, FF, matrix(C, [0 1; 1 0])));
 julia> phi = gmodule(G, im);
 
 julia> character(phi)
-class_function(character table of G, [2, 0, -1, 2, 0])
+class_function(character table of G, QQAbElem{AbsSimpleNumFieldElem}[2, 0, -1, 2, 0])
 
 julia> schur_index(ans)
 1
@@ -71,7 +71,7 @@ julia> schur_index(ans)
 julia> T = matrix(C, [C(1) -z; z+1 z]);
 
 julia> [matrix(x)^T for x in action(phi)]
-4-element Vector{AbstractAlgebra.Generic.MatSpaceElem{QQAbFieldElem{AbsSimpleNumFieldElem}}}:
+4-element Vector{AbstractAlgebra.Generic.MatSpaceElem{QQAbElem{AbsSimpleNumFieldElem}}}:
  [1 0; -1 -1]
  [0 1; -1 -1]
  [1 0; 0 1]
diff --git a/test/book/cornerstones/number-theory/galoismod.jlcon b/test/book/cornerstones/number-theory/galoismod.jlcon
index 64c37466576c..6e46fdcb4463 100644
--- a/test/book/cornerstones/number-theory/galoismod.jlcon
+++ b/test/book/cornerstones/number-theory/galoismod.jlcon
@@ -55,18 +55,18 @@ julia> V, f = galois_module(K); OK = ring_of_integers(K); M = f(OK);
 julia> fl, c = is_free_with_basis(M); # the elements of c are a basis
 
 julia> b = preimage(f, c[1]) # the element might different per session
-a^3
+a
 
 julia> A, mA = automorphism_group(K);
 
 julia> X = matrix(ZZ, [coordinates(OK(mA(s)(b))) for s in A])
+[0    1    0    0]
 [0    0    0    1]
 [1   -1    1   -1]
 [0    0   -1    0]
-[0    1    0    0]
 
 julia> det(X)
--1
+1
 
 julia> k, = number_field(x^4 - 13*x^2 + 16); candidates = []; for F in abelian_normal_extensions(k, [2], ZZ(10)^13)
          K, = absolute_simple_field(number_field(F))
diff --git a/test/book/cornerstones/number-theory/galoismod_1.jl b/test/book/cornerstones/number-theory/galoismod_1.jlcon
similarity index 100%
rename from test/book/cornerstones/number-theory/galoismod_1.jl
rename to test/book/cornerstones/number-theory/galoismod_1.jlcon
diff --git a/test/book/cornerstones/number-theory/galoismod_2.jl b/test/book/cornerstones/number-theory/galoismod_2.jlcon
similarity index 100%
rename from test/book/cornerstones/number-theory/galoismod_2.jl
rename to test/book/cornerstones/number-theory/galoismod_2.jlcon
diff --git a/test/book/cornerstones/number-theory/intro.jlcon b/test/book/cornerstones/number-theory/intro.jlcon
index 55237aeef016..8a7bda27db33 100644
--- a/test/book/cornerstones/number-theory/intro.jlcon
+++ b/test/book/cornerstones/number-theory/intro.jlcon
@@ -39,13 +39,7 @@ julia> discriminant(OK)
 julia> prime_ideals_over(OK, 7)
 2-element Vector{AbsSimpleNumFieldOrderIdeal}:
  <7, a + 5>
-Norm: 7
-Minimum: 7
-two normal wrt: 7
  <7, a + 2>
-Norm: 7
-Minimum: 7
-two normal wrt: 7
 
 julia> factor(change_coefficient_ring(GF(7), x^2 - 235))
 1 * (x + 5) * (x + 2)
@@ -62,16 +56,11 @@ Z/6
 
 julia> m(zero(A)) # apply m to the neutral element of A
 <1, 1>
-Norm: 1
-Minimum: 1
-principal generator 1
-two normal wrt: 1
+[...]
 
 julia> P = prime_ideals_over(OK, 2)[1]
 <2, a + 1>
-Norm: 2
-Minimum: 2
-two normal wrt: 2
+[...]
 
 julia> preimage(m, P)
 Abelian group element [3]
@@ -94,4 +83,4 @@ julia> preimage(mU, -214841715*a - 3293461126)
 Abelian group element [1, 5]
 
 julia> -214841715*a - 3293461126 == (-1)^1 * (3a + 46)^5
-true
\ No newline at end of file
+true
diff --git a/test/book/cornerstones/number-theory/sym_1.jl b/test/book/cornerstones/number-theory/sym_1.jlcon
similarity index 100%
rename from test/book/cornerstones/number-theory/sym_1.jl
rename to test/book/cornerstones/number-theory/sym_1.jlcon
diff --git a/test/book/cornerstones/number-theory/unit_log_plot.jl b/test/book/cornerstones/number-theory/unit_log_plot.jlcon
similarity index 100%
rename from test/book/cornerstones/number-theory/unit_log_plot.jl
rename to test/book/cornerstones/number-theory/unit_log_plot.jlcon
diff --git a/test/book/cornerstones/polyhedral-geometry/D222Computation.jlcon b/test/book/cornerstones/polyhedral-geometry/D222Computation.jlcon
index 76949e796f79..8d187fd581c5 100644
--- a/test/book/cornerstones/polyhedral-geometry/D222Computation.jlcon
+++ b/test/book/cornerstones/polyhedral-geometry/D222Computation.jlcon
@@ -4,9 +4,12 @@ julia> R, x, c = polynomial_ring(QQ, :x=>1:3, :c=>(0:1,0:1,0:1));
 
 julia> f = sum(prod(x[i]^Int(p[i]) for i=1:3)
                * c[(Vector{Int}(p)+[1,1,1])...] for p=lattice_points(C))
-x[1]*x[2]*x[3]*c[1, 1, 1] + x[1]*x[2]*c[1, 1, 0] + x[1]*x[3]*c[1, 0, 1] + x[1]*c[1, 0, 0] + x[2]*x[3]*c[0, 1, 1] + x[2]*c[0, 1, 0] + x[3]*c[0, 0, 1] + c[0, 0, 0]
+x[1]*x[2]*x[3]*c[1, 1, 1] + x[1]*x[2]*c[1, 1, 0] + 
+x[1]*x[3]*c[1, 0, 1] + x[1]*c[1, 0, 0] + 
+x[2]*x[3]*c[0, 1, 1] + x[2]*c[0, 1, 0] + 
+x[3]*c[0, 0, 1] + c[0, 0, 0]
 
 julia> I = ideal(R, vcat([derivative(f,t) for t = x], [f]));
 
 julia> D222 = eliminate(I,x)[1]
-c[0, 0, 0]^2*c[1, 1, 1]^2 - 2*c[0, 0, 0]*c[1, 0, 0]*c[0, 1, 1]*c[1, 1, 1] - 2*c[0, 0, 0]*c[0, 1, 0]*c[1, 0, 1]*c[1, 1, 1] - 2*c[0, 0, 0]*c[1, 1, 0]*c[0, 0, 1]*c[1, 1, 1] + 4*c[0, 0, 0]*c[1, 1, 0]*c[1, 0, 1]*c[0, 1, 1] + c[1, 0, 0]^2*c[0, 1, 1]^2 + 4*c[1, 0, 0]*c[0, 1, 0]*c[0, 0, 1]*c[1, 1, 1] - 2*c[1, 0, 0]*c[0, 1, 0]*c[1, 0, 1]*c[0, 1, 1] - 2*c[1, 0, 0]*c[1, 1, 0]*c[0, 0, 1]*c[0, 1, 1] + c[0, 1, 0]^2*c[1, 0, 1]^2 - 2*c[0, 1, 0]*c[1, 1, 0]*c[0, 0, 1]*c[1, 0, 1] + c[1, 1, 0]^2*c[0, 0, 1]^2
+c[0, 0, 0]^2*c[1, 1, 1]^2 - 2*c[0, 0, 0]*c[1, 0, 0]*c[0, 1, 1]*c[1, 1, 1] - [...] + c[1, 1, 0]^2*c[0, 0, 1]^2
diff --git a/test/book/cornerstones/polyhedral-geometry/GKZ_orbits.jlcon b/test/book/cornerstones/polyhedral-geometry/GKZ_orbits.jlcon
index 1524f987593e..6d47634cc605 100644
--- a/test/book/cornerstones/polyhedral-geometry/GKZ_orbits.jlcon
+++ b/test/book/cornerstones/polyhedral-geometry/GKZ_orbits.jlcon
@@ -15,8 +15,7 @@ julia> OrbitRepresentatives=
 julia> OrbitSizes = 
         [length(
           filter(x->minimum(gset(G,permuted,[x]))==u,
-          GKZ_Vectors)
-         ) for u in OrbitRepresentatives];
+          GKZ_Vectors)) for u in OrbitRepresentatives];
 
 julia> show(OrbitSizes)
 [12, 24, 24, 8, 2, 4]
diff --git a/test/book/cornerstones/polyhedral-geometry/g-vector-example.jlcon b/test/book/cornerstones/polyhedral-geometry/g-vector-example.jlcon
index 192fe9c80752..1ac2ceeb3ad1 100644
--- a/test/book/cornerstones/polyhedral-geometry/g-vector-example.jlcon
+++ b/test/book/cornerstones/polyhedral-geometry/g-vector-example.jlcon
@@ -3,7 +3,9 @@ Polytope in ambient dimension 6
 
 julia> show(f_vector(P))
 ZZRingElem[30, 336, 1468, 2874, 2568, 856]
+
 julia> show(h_vector(P))
 ZZRingElem[1, 24, 201, 404, 201, 24, 1]
+
 julia> show(g_vector(P))
 ZZRingElem[1, 23, 177, 203]
diff --git a/test/book/cornerstones/polyhedral-geometry/g-vectors-upper-bound.jlcon b/test/book/cornerstones/polyhedral-geometry/g-vectors-upper-bound.jlcon
index 38deeb26b927..95f007329a82 100644
--- a/test/book/cornerstones/polyhedral-geometry/g-vectors-upper-bound.jlcon
+++ b/test/book/cornerstones/polyhedral-geometry/g-vectors-upper-bound.jlcon
@@ -6,6 +6,7 @@ julia> max_g2 = maximum(g_vectors[:,1])
 
 julia> show(upper_bound_g_vector(6,30))
 [1, 23, 276, 2300]
+
 julia> ub = [ Int(Polymake.polytope.pseudopower(g2,2)) for g2 in min_g2:max_g2 ]
 34-element Vector{Int64}:
   990
diff --git a/test/book/cornerstones/polyhedral-geometry/lp.jlcon b/test/book/cornerstones/polyhedral-geometry/lp.jlcon
index 9936ea50dcf4..3e448c0e0b09 100644
--- a/test/book/cornerstones/polyhedral-geometry/lp.jlcon
+++ b/test/book/cornerstones/polyhedral-geometry/lp.jlcon
@@ -5,11 +5,11 @@ julia> LP = linear_program(P, [1,1], convention=:max)
 Linear program
    max{c*x + k | x in P}
 where P is a Polyhedron{QQFieldElem} and
-   c=QQFieldElem[1, 1]
+   c=Polymake.LibPolymake.Rational[1 1]
    k=0
 
 julia> optimal_value(LP)
-100//11
+100/11
 
 julia> optimal_vertex(LP)
 2-element PointVector{QQFieldElem}:
diff --git a/test/book/cornerstones/polyhedral-geometry/not-pointed.jlcon b/test/book/cornerstones/polyhedral-geometry/not-pointed.jlcon
index 58bf9c305357..5c0317e19c89 100644
--- a/test/book/cornerstones/polyhedral-geometry/not-pointed.jlcon
+++ b/test/book/cornerstones/polyhedral-geometry/not-pointed.jlcon
@@ -5,4 +5,5 @@ julia> vertices(Q)
 0-element SubObjectIterator{PointVector{QQFieldElem}}
 
 julia> minimal_faces(Q)
-(base_points = PointVector{QQFieldElem}[[0, 0, 0]], lineality_basis = RayVector{QQFieldElem}[[1, 0, 0]])
+(base_points = PointVector{QQFieldElem}[[0, 0, 0]], 
+ lineality_basis = RayVector{QQFieldElem}[[1, 0, 0]])
diff --git a/test/book/cornerstones/polyhedral-geometry/pentagon.jlcon b/test/book/cornerstones/polyhedral-geometry/pentagon.jlcon
index cf96a5f18ba5..7d2aefecf1e5 100644
--- a/test/book/cornerstones/polyhedral-geometry/pentagon.jlcon
+++ b/test/book/cornerstones/polyhedral-geometry/pentagon.jlcon
@@ -11,7 +11,6 @@ x_1 - 2*x_2 <= 1
 x_1 <= 1
 x_2 <= 1
 
-
 julia> f_vector(P)
 2-element Vector{ZZRingElem}:
  5
diff --git a/test/book/cornerstones/polyhedral-geometry/platonic.jlcon b/test/book/cornerstones/polyhedral-geometry/platonic.jlcon
index 5be3803224d1..f216bce0e5cd 100644
--- a/test/book/cornerstones/polyhedral-geometry/platonic.jlcon
+++ b/test/book/cornerstones/polyhedral-geometry/platonic.jlcon
@@ -1,4 +1,5 @@
-julia> [ f_vector(P) for P in [T, cube(3), cross_polytope(3), dodecahedron(), icosahedron()] ]
+julia> [ f_vector(P) for P in [T, cube(3), cross_polytope(3),
+                               dodecahedron(), icosahedron()] ]
 5-element Vector{Vector{ZZRingElem}}:
  [4, 6, 4]
  [8, 12, 6]
diff --git a/test/book/ordered_examples.json b/test/book/ordered_examples.json
index 9100288ceb77..168f6562705d 100644
--- a/test/book/ordered_examples.json
+++ b/test/book/ordered_examples.json
@@ -2,86 +2,15 @@
   "_ns": {
     "Oscar": [
       "https://github.com/oscar-system/Oscar.jl",
-      "1.0.0"
+      "1.1.1"
     ]
   },
   "_type": {
     "name": "Dict",
     "params": {
       "key_type": "String",
-      "specialized/boehm-breuer-git-fans": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "specialized/markwig-ristau-schleis-faithful-tropicalization": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "specialized/holt-ren-tropical-geometry": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "specialized/bies-turner-string-theory-applications": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "specialized/aga-boehm-hoffmann-markwig-traore": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "specialized/joswig-kastner-lorenz-confirmable-workflows": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "cornerstones/number-theory": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "cornerstones/groups": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "introduction/introduction": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "specialized/eder-mohr-ideal-theoretic": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "specialized/kuehne-schroeter-matroids": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "cornerstones/polyhedral-geometry": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "specialized/rose-sturmfels-telen-tropical-implicitization": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "specialized/fang-fourier-monomial-bases": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "specialized/brandhorst-zach-fibration-hopping": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "specialized/breuer-nebe-parker-orthogonal-discriminants": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "cornerstones/algebraic-geometry": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "specialized/decker-schmitt-invariant-theory": {
-        "name": "Vector",
-        "params": "String"
-      },
-      "specialized/bies-kastner-toric-geometry": {
+      "value_type": "Vector",
+      "value_params": {
         "name": "Vector",
         "params": "String"
       }
@@ -144,16 +73,16 @@
       "intro.jlcon",
       "intro_plot_lattice.jlcon",
       "unit_plot.jlcon",
-      "unit_log_plot.jl",
+      "unit_log_plot.jlcon",
       "general.jlcon",
       "embeddings.jlcon",
       "intro4.jlcon",
       "sym.jlcon",
-      "sym_1.jl",
+      "sym_1.jlcon",
       "cohenlenstra.jlcon",
       "galoismod.jlcon",
-      "galoismod_1.jl",
-      "galoismod_2.jl"
+      "galoismod_1.jlcon",
+      "galoismod_2.jlcon"
     ],
     "cornerstones/groups": [
       "intro.jlcon",
@@ -220,13 +149,16 @@
       "chow_fan.jlcon",
       "chow_transl.jlcon"
     ],
-    "specialized/fang-fourier-monomial-bases": [
+    "specialized/flake-fourier-monomial-bases": [
       "get-operators.jlcon",
       "basis.jlcon",
       "fflv.jlcon",
       "string.jlcon",
       "lusztig.jlcon",
-      "nz.jlcon"
+      "nz.jlcon",
+      "sl7-cases.jlcon",
+      "oscar-vs-gap.jlcon",
+      "gap.jlcon"
     ],
     "specialized/brandhorst-zach-fibration-hopping": [
       "vinberg_1.jlcon",
diff --git a/test/book/specialized/bies-kastner-toric-geometry/p1xp1_cohomologies.jlcon b/test/book/specialized/bies-kastner-toric-geometry/p1xp1_cohomologies.jlcon
index 2971bee5b265..c6da260e5df0 100644
--- a/test/book/specialized/bies-kastner-toric-geometry/p1xp1_cohomologies.jlcon
+++ b/test/book/specialized/bies-kastner-toric-geometry/p1xp1_cohomologies.jlcon
@@ -1,4 +1,4 @@
-julia> P1 = projective_space(NormalToricVariety,1)
+julia> P1 = projective_space(NormalToricVariety, 1)
 Normal toric variety
 
 julia> l = toric_line_bundle(P1*P1, [-3,1])
diff --git a/test/book/specialized/bies-kastner-toric-geometry/p1xp1_vanishing_sets.jlcon b/test/book/specialized/bies-kastner-toric-geometry/p1xp1_vanishing_sets.jlcon
index 4d412ab3c970..2cad9f47490e 100644
--- a/test/book/specialized/bies-kastner-toric-geometry/p1xp1_vanishing_sets.jlcon
+++ b/test/book/specialized/bies-kastner-toric-geometry/p1xp1_vanishing_sets.jlcon
@@ -1,4 +1,4 @@
-julia> P1 = projective_space(NormalToricVariety,1)
+julia> P1 = projective_space(NormalToricVariety, 1)
 Normal toric variety
 
 julia> v0, v1, v2 = vanishing_sets(P1*P1)
diff --git a/test/book/specialized/bies-kastner-toric-geometry/polyhedral_fan.jlcon b/test/book/specialized/bies-kastner-toric-geometry/polyhedral_fan.jlcon
index b9ebde553b16..d8a4134ddece 100644
--- a/test/book/specialized/bies-kastner-toric-geometry/polyhedral_fan.jlcon
+++ b/test/book/specialized/bies-kastner-toric-geometry/polyhedral_fan.jlcon
@@ -18,6 +18,5 @@ julia> maximal_cones(IncidenceMatrix, tv)
 [2, 3]
 [2, 4]
 
-
 julia> Sigma = polyhedral_fan(tv)
 Polyhedral fan in ambient dimension 2
diff --git a/test/book/specialized/bies-kastner-toric-geometry/starsubdivision.jlcon b/test/book/specialized/bies-kastner-toric-geometry/starsubdivision.jlcon
index d576e96ecbc0..b7aa7131a187 100644
--- a/test/book/specialized/bies-kastner-toric-geometry/starsubdivision.jlcon
+++ b/test/book/specialized/bies-kastner-toric-geometry/starsubdivision.jlcon
@@ -19,7 +19,7 @@ julia> rays(maximal_cones(v)[1])
  [-1, 1]
 
 julia> bu = blow_up(v, [0,1])
-Toric blowup morphism
+Toric blowdown morphism
 
 julia> v2 = domain(bu)
 Normal toric variety
@@ -31,7 +31,7 @@ julia> is_affine(v2)
 false
 
 julia> cox_ring(v2)
-Multivariate polynomial ring in 3 variables over QQ graded by
+Multivariate polynomial ring in 3 variables over QQ graded by 
   u1 -> [1]
   u2 -> [1]
   e -> [-2]
diff --git a/test/book/specialized/brandhorst-zach-fibration-hopping/vinberg_1.jlcon b/test/book/specialized/brandhorst-zach-fibration-hopping/vinberg_1.jlcon
index fcfbfc80f870..2324e16baaa0 100644
--- a/test/book/specialized/brandhorst-zach-fibration-hopping/vinberg_1.jlcon
+++ b/test/book/specialized/brandhorst-zach-fibration-hopping/vinberg_1.jlcon
@@ -40,7 +40,7 @@ julia> length(Xchambers)
 1
 
 julia> C = Xchambers[1]
-Chamber in dimension 20 with 36 walls
+Chamber  in dimension 20 with 36 walls
 
 julia> Xelliptic_classes = [f for f in rays(C) if 0 == f*gram_matrix(NS)*transpose(f)];
 
diff --git a/test/book/specialized/brandhorst-zach-fibration-hopping/vinberg_2.jlcon b/test/book/specialized/brandhorst-zach-fibration-hopping/vinberg_2.jlcon
index e36b701630ba..43a95ef72d16 100644
--- a/test/book/specialized/brandhorst-zach-fibration-hopping/vinberg_2.jlcon
+++ b/test/book/specialized/brandhorst-zach-fibration-hopping/vinberg_2.jlcon
@@ -50,33 +50,14 @@ with default covering
     6: [(z//y), (x//y), s]
 
 julia> piS = weierstrass_contraction(Y1)
-Composite morphism of
-  Hom: elliptic surface with generic fiber -x^3 + y^2 - t^7 + 2*t^6 - t^5 -> scheme over QQ covered with 44 patches
-  Hom: scheme over QQ covered with 44 patches -> scheme over QQ covered with 40 patches
-  Hom: scheme over QQ covered with 40 patches -> scheme over QQ covered with 38 patches
-  Hom: scheme over QQ covered with 38 patches -> scheme over QQ covered with 36 patches
-  Hom: scheme over QQ covered with 36 patches -> scheme over QQ covered with 32 patches
-  Hom: scheme over QQ covered with 32 patches -> scheme over QQ covered with 30 patches
-  Hom: scheme over QQ covered with 30 patches -> scheme over QQ covered with 28 patches
-  Hom: scheme over QQ covered with 28 patches -> scheme over QQ covered with 26 patches
-  Hom: scheme over QQ covered with 26 patches -> scheme over QQ covered with 24 patches
-  Hom: scheme over QQ covered with 24 patches -> scheme over QQ covered with 22 patches
-  Hom: scheme over QQ covered with 22 patches -> scheme over QQ covered with 20 patches
-  Hom: scheme over QQ covered with 20 patches -> scheme over QQ covered with 18 patches
-  Hom: scheme over QQ covered with 18 patches -> scheme over QQ covered with 16 patches
-  Hom: scheme over QQ covered with 16 patches -> scheme over QQ covered with 14 patches
-  Hom: scheme over QQ covered with 14 patches -> scheme over QQ covered with 12 patches
-  Hom: scheme over QQ covered with 12 patches -> scheme over QQ covered with 10 patches
-  Hom: scheme over QQ covered with 10 patches -> scheme over QQ covered with 6 patches
-
-
+Composite morphism of[...]
 
 julia> basisNSY1, gramTriv = trivial_lattice(Y1);
 
 julia> [(i[1],i[2]) for i in reducible_fibers(Y1)]
 3-element Vector{Tuple{Vector{QQFieldElem}, Tuple{Symbol, Int64}}}:
- ([0, 1], (:E, 8))
  ([1, 1], (:A, 2))
+ ([0, 1], (:E, 8))
  ([1, 0], (:E, 8))
 
 julia> basisNSY1, _, NSY1 = algebraic_lattice(Y1);
@@ -85,6 +66,8 @@ julia> basisNSY1
 20-element Vector{Any}:
  Fiber over (2, 1)
  section: (0 : 1 : 0)
+ component A2_1 of fiber over (1, 1)
+ component A2_2 of fiber over (1, 1)
  component E8_1 of fiber over (0, 1)
  component E8_2 of fiber over (0, 1)
  component E8_3 of fiber over (0, 1)
@@ -93,8 +76,6 @@ julia> basisNSY1
  component E8_6 of fiber over (0, 1)
  component E8_7 of fiber over (0, 1)
  component E8_8 of fiber over (0, 1)
- component A2_1 of fiber over (1, 1)
- component A2_2 of fiber over (1, 1)
  component E8_1 of fiber over (1, 0)
  component E8_2 of fiber over (1, 0)
  component E8_3 of fiber over (1, 0)
@@ -107,16 +88,31 @@ julia> basisNSY1
 julia> basisNSY1[1]
 Effective weil divisor Fiber over (2, 1)
   on elliptic surface with generic fiber -x^3 + y^2 - t^7 + 2*t^6 - t^5
-with coefficients in integer ring
+with coefficients in integer Ring
 given as the formal sum of
   1 * sheaf of ideals
 
 julia> b, I = Oscar._is_equal_up_to_permutation_with_permutation(gram_matrix(NS), gram_matrix(NSY1))
-(true, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 20, 11, 12, 13, 14, 15, 16, 17, 18])
+(true, [1, 2, 19, 20, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18])
 
 julia> @assert gram_matrix(NSY1) == gram_matrix(NS)[I,I]
 
-julia> Oscar.horizontal_decomposition(Y1, fibers[2][I])[2];
+julia> Oscar.horizontal_decomposition(Y1, fibers[2][I])[2]
+Effective weil divisor
+  on elliptic surface with generic fiber -x^3 + y^2 - t^7 + 2*t^6 - t^5
+with coefficients in integer Ring
+given as the formal sum of
+  2 * component E8_7 of fiber over (0, 1)
+  2 * section: (0 : 1 : 0)
+  1 * component A2_0 of fiber over (1, 1)
+  1 * component E8_0 of fiber over (1, 0)
+  2 * component E8_4 of fiber over (0, 1)
+  2 * component E8_3 of fiber over (0, 1)
+  1 * component E8_2 of fiber over (0, 1)
+  2 * component E8_0 of fiber over (0, 1)
+  2 * component E8_6 of fiber over (0, 1)
+  2 * component E8_5 of fiber over (0, 1)
+  1 * component E8_8 of fiber over (0, 1)
 
 julia> representative(elliptic_parameter(Y1, fibers[2][I]))
 (x//z)//(t^3 - t^2)
@@ -131,15 +127,17 @@ Elliptic surface
   over rational field
 with generic fiber
   -x^3 + t^3*x^2 - t^3*x + y^2
+and relatively minimal model
+  scheme over QQ covered with 47 patches
 
-julia> E2 = generic_fiber(Y2); tt = gen(kt2);
+julia> E2 = generic_fiber(Y2); t2 = gen(kt2);
 
-julia> P2 = E2([tt^3, tt^3]); set_mordell_weil_basis!(Y2, [P2]);
+julia> P2 = E2([t2^3, t2^3]); set_mordell_weil_basis!(Y2, [P2]);
 
 julia> U2 = weierstrass_chart_on_minimal_model(Y2); U1 = weierstrass_chart_on_minimal_model(Y1);
 
 julia> imgs = phi2.(phi1.(ambient_coordinates(U1))) # k(Y1) -> k(Y2)
-3-element Vector{AbstractAlgebra.Generic.FracFieldElem{QQMPolyRingElem}}:
+3-element Vector{AbstractAlgebra.Generic.Frac{QQMPolyRingElem}}:
  (-(x//z)*t + t)//(x//z)^3
  ((x//z)*(y//z) - (y//z))//(x//z)^5
  1//(x//z)
@@ -151,6 +149,27 @@ julia> set_attribute!(phi_rat, :is_isomorphism=>true);
 julia> pullbackDivY1 = [pullback(phi_rat, D) for D in basisNSY1];
 
 julia> B = [basis_representation(Y2, D) for D in pullbackDivY1];
+20-element Vector{Vector{QQFieldElem}}:
+ [4, 2, 0, 0, 0, 0, 0, 0, 0, -4, -4, -8, -7, -6, -5, -4, -3, -2, -1, 0]
+ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
+ [4, 2, -1, -2, -3, -5//2, -2, -3//2, -3//2, -3, -5//2, -5, -9//2, -4, -7//2, -3, -5//2, -2, -3//2, -1]
+ [0, 0, 1, 2, 3, 5//2, 2, 3//2, 3//2, -2, -3//2, -3, -5//2, -2, -3//2, -1, -1//2, 0, 1//2, 1]
+ [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+ [1, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -2, -2, -2, -2, -2, -2, -2, -1, 0]
+ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0]
+ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]
+ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0]
+ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0]
+ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
+ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]
+ [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+ [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+ [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+ [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+ [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+ [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+ [2, 1, -1, -2, -3, -5//2, -2, -3//2, -3//2, -2, -5//2, -4, -7//2, -3, -5//2, -2, -3//2, -1, -1//2, 0]
+ [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
 
 julia> B = matrix(QQ, 20, 20, reduce(vcat, B)); NSY2 = algebraic_lattice(Y2)[3];
 
@@ -158,57 +177,17 @@ julia> NSY1inY2 = lattice(ambient_space(NSY2),B);
 
 julia> @assert NSY1inY2 == NSY2 && gram_matrix(NSY1inY2) == gram_matrix(NSY1)
 
-julia> fibers_in_Y2 = [f[I]*B for f in fibers];
+julia> fibers_in_Y2 = [f[I]*B for f in fibers]
+6-element Vector{Vector{QQFieldElem}}:
+ [4, 2, 0, 0, 0, 0, 0, 0, 0, -4, -4, -8, -7, -6, -5, -4, -3, -2, -1, 0]
+ [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+ [5, 2, -2, -3, -4, -3, -2, -1, -2, -5, -4, -8, -7, -6, -5, -4, -3, -2, -1, 0]
+ [4, 2, -2, -4, -6, -9//2, -3, -3//2, -7//2, -3, -5//2, -5, -9//2, -4, -7//2, -3, -5//2, -2, -3//2, 0]
+ [2, 1, -1, -2, -3, -2, -1, 0, -2, -1, -1, -2, -2, -2, -2, -2, -2, -1, 0, 1]
+ [2, 1, 0, 0, 0, 0, 0, 0, 0, -2, -2, -4, -4, -4, -4, -3, -2, -1, 0, 1]
 
 julia> f3 = fibers[3][I]; representative(elliptic_parameter(Y1, f3))
 (x//z)//t^2
 
 julia> g3a, phi3a = two_neighbor_step(Y1, f3); g3a
 -x^3 + (-t^3 + 2)*x^2 - x + y^2
-
-julia> @assert all(inner_product(ambient_space(NSY2), i,i) == 0 for i in fibers_in_Y2)
-
-julia> [representative(elliptic_parameter(Y2, f)) for f in fibers_in_Y2[4:6]]
-3-element Vector{AbstractAlgebra.Generic.FracFieldElem{QQMPolyRingElem}}:
- (y//z)//((x//z)*t)
- ((y//z) + t^3)//((x//z)*t - t^4)
- ((y//z) + t^3)//((x//z) - t^3)
-
-julia> g,_ = two_neighbor_step(Y2, fibers_in_Y2[4]);g
--1//4*x^4 - 1//2*t^2*x^3 - 1//4*t^4*x^2 + x + y^2
-
-julia> R = parent(g); K_t = base_ring(R);
-
-julia> (x,y) = gens(R); P = K_t.([0,0]); # rational point
-
-julia> g, _ = transform_to_weierstrass(g, x, y, P);
-
-julia> E4 = elliptic_curve(g, x, y)
-Elliptic curve with equation
-y^2 = x^3 + 1//4*t^4*x^2 - 1//2*t^2*x + 1//4
-
-julia> g,_ = two_neighbor_step(Y2, fibers_in_Y2[5]);g
-t^2*x^3 + (-1//4*t^4 + 2*t)*x^2 + x + y^2
-
-julia> R = parent(g); K_t = base_ring(R);
-
-julia> (x,y) = gens(R); P = K_t.([0,0]); # rational point
-
-julia> g, _ = transform_to_weierstrass(g, x, y, P);
-
-julia> E5 = elliptic_curve(g, x, y)
-Elliptic curve with equation
-y^2 = x^3 + (1//4*t^4 - 2*t)*x^2 + t^2*x
-
-julia> g,_ = two_neighbor_step(Y2, fibers_in_Y2[6]);g
-(t^2 + 2*t + 1)*x^3 + y^2 - 1//4*t^4
-
-julia> R = parent(g); K_t = base_ring(R); t = gen(K_t);
-
-julia> (x,y) = gens(R); P = K_t.([0,1//2*t^2]); # rational point
-
-julia> g, _ = transform_to_weierstrass(g, x, y, P);
-
-julia> E6 = elliptic_curve(g, x, y)
-Elliptic curve with equation
-y^2 + (-t^2 - 2*t - 1)//t^4*y = x^3
diff --git a/test/book/specialized/breuer-nebe-parker-orthogonal-discriminants/expl_G23_tbl.jlcon b/test/book/specialized/breuer-nebe-parker-orthogonal-discriminants/expl_G23_tbl.jlcon
index 79c5af782859..3a9caef550a6 100644
--- a/test/book/specialized/breuer-nebe-parker-orthogonal-discriminants/expl_G23_tbl.jlcon
+++ b/test/book/specialized/breuer-nebe-parker-orthogonal-discriminants/expl_G23_tbl.jlcon
@@ -1,18 +1,18 @@
 julia> Oscar.OrthogonalDiscriminants.show_with_ODs(
-character_table("G2(3)", 2))
+       character_table("G2(3)", 2))
 G2(3)mod2
 
-     2   6   3   3   .  1  1  .  .  .  .   .   .
-     3   6   6   6   6  4  4  .  3  3  3   .   .
-     7   1   .   .   .  .  .  1  .  .  .   .   .
-    13   1   .   .   .  .  .  .  .  .  .   1   1
-      
- 1a  3a  3b  3c 3d 3e 7a 9a 9b 9c 13a 13b
-    2P  1a  3a  3b  3c 3d 3e 7a 9a 9c 9b 13b 13a
-    3P  1a  1a  1a  1a 1a 1a 7a 3c 3c 3c 13a 13b
-    7P  1a  3a  3b  3c 3d 3e 1a 9a 9b 9c 13b 13a
-   13P  1a  3a  3b  3c 3d 3e 7a 9a 9b 9c  1a  1a
-     d OD   2                                          
+            2   6   3   3   .  1  1  .  .  .  .   .   .
+            3   6   6   6   6  4  4  .  3  3  3   .   .
+            7   1   .   .   .  .  .  1  .  .  .   .   .
+           13   1   .   .   .  .  .  .  .  .  .   1   1
+
+               1a  3a  3b  3c 3d 3e 7a 9a 9b 9c 13a 13b
+           2P  1a  3a  3b  3c 3d 3e 7a 9a 9c 9b 13b 13a
+           3P  1a  1a  1a  1a 1a 1a 7a 3c 3c 3c 13a 13b
+           7P  1a  3a  3b  3c 3d 3e 1a 9a 9b 9c 13b 13a
+          13P  1a  3a  3b  3c 3d 3e 7a 9a 9b 9c  1a  1a
+     d OD   2
  X_1 1      +   1   1   1   1  1  1  1  1  1  1   1   1
  X_2 1 O-   +  14   5   5  -4  2 -1  .  2 -1 -1   1   1
  X_3 2      o  64  -8  -8   1  4 -2  1  1  A /A  -1  -1
@@ -29,4 +29,4 @@ X_12 1 O+   + 832 -32 -32  -5  4  4 -1  1  1  1   .   .
 A = 3z_3 + 1
 /A = -3z_3 - 2
 B = -z_13^11 - z_13^8 - z_13^7 - z_13^6 - z_13^5 - z_13^2 - 1
-B* = z_13^11 + z_13^8 + z_13^7 + z_13^6 + z_13^5 + z_13^2
\ No newline at end of file
+B* = z_13^11 + z_13^8 + z_13^7 + z_13^6 + z_13^5 + z_13^2
diff --git a/test/book/specialized/decker-schmitt-invariant-theory/H3.jlcon b/test/book/specialized/decker-schmitt-invariant-theory/H3.jlcon
index 3c7ed37b97b5..9e51b1a925be 100644
--- a/test/book/specialized/decker-schmitt-invariant-theory/H3.jlcon
+++ b/test/book/specialized/decker-schmitt-invariant-theory/H3.jlcon
@@ -45,7 +45,7 @@ julia> length(MP)
 
 julia> [[ kernel(reduce(hcat, [ K[coeff(p, x); coeff(p, y); coeff(p, z)] for p in gens(Q) ])) for Q in MP ]]
 1-element Vector{Vector{AbstractAlgebra.Generic.MatSpaceElem{AbsSimpleNumFieldElem}}}:
- [[a+1 0 1], [-a 0 1], [-1 0 1], [0 a+1 1], [0 -a 1], [0 -1 1], [-a 1 0], [a+1 1 0], [-1 1 0]]
+ [[a+1 1 0], [-a 1 0], [-1 1 0], [a+1 0 1], [-a 0 1], [-1 0 1], [0 -a 1], [0 a+1 1], [0 -1 1]]
 
 julia> T, t = polynomial_ring(QQ, "t")
 (Univariate polynomial ring in t over QQ, t)
diff --git a/test/book/specialized/decker-schmitt-invariant-theory/cox_ring.jlcon b/test/book/specialized/decker-schmitt-invariant-theory/cox_ring.jlcon
index a4aec0d11cd2..f66cc90d9a35 100644
--- a/test/book/specialized/decker-schmitt-invariant-theory/cox_ring.jlcon
+++ b/test/book/specialized/decker-schmitt-invariant-theory/cox_ring.jlcon
@@ -32,9 +32,9 @@ julia> A, AtoR = affine_algebra(RH)
 
 julia> map(AtoR, gens(A))
 3-element Vector{MPolyDecRingElem{AbsSimpleNumFieldElem, AbstractAlgebra.Generic.MPoly{AbsSimpleNumFieldElem}}}:
- x[1]^2
- x[2]^2
  x[3]^2
+ x[2]^2
+ x[1]^2
 
 julia> VG = linear_quotient(G)
 Linear quotient by matrix group of degree 3 over K
@@ -44,9 +44,9 @@ julia> B, BtoR = cox_ring(VG)
 
 julia> map(BtoR, gens(B))
 3-element Vector{MPolyDecRingElem{AbsSimpleNumFieldElem, AbstractAlgebra.Generic.MPoly{AbsSimpleNumFieldElem}}}:
- z_3*x[1]^2 + (-z_3 - 1)*x[2]^2 + x[3]^2
+ x[1]^2 + z_3*x[2]^2 + (-z_3 - 1)*x[3]^2
  x[1]^2 + x[2]^2 + x[3]^2
- (-z_3 - 1)*x[1]^2 + z_3*x[2]^2 + x[3]^2
+ x[1]^2 + (-z_3 - 1)*x[2]^2 + z_3*x[3]^2
 
 julia> f1 = BtoR(B[1]); f2 = BtoR(B[2]); f3 = BtoR(B[3]);
 
diff --git a/test/book/specialized/decker-schmitt-invariant-theory/gleason.jlcon b/test/book/specialized/decker-schmitt-invariant-theory/gleason.jlcon
index 76646d473ba4..04bbe290781b 100644
--- a/test/book/specialized/decker-schmitt-invariant-theory/gleason.jlcon
+++ b/test/book/specialized/decker-schmitt-invariant-theory/gleason.jlcon
@@ -20,7 +20,7 @@ julia> RG = invariant_ring(G);
 julia> f = fundamental_invariants(RG)
 2-element Vector{MPolyDecRingElem{AbsNonSimpleNumFieldElem, AbstractAlgebra.Generic.MPoly{AbsNonSimpleNumFieldElem}}}:
  x[1]^8 + 14*x[1]^4*x[2]^4 + x[2]^8
- x[1]^20*x[2]^4 - 4*x[1]^16*x[2]^8 + 6*x[1]^12*x[2]^12 - 4*x[1]^8*x[2]^16 + x[1]^4*x[2]^20
+ x[1]^24 + 10626//1025*x[1]^20*x[2]^4 + 735471//1025*x[1]^16*x[2]^8 + 2704156//1025*x[1]^12*x[2]^12 + 735471//1025*x[1]^8*x[2]^16 + 10626//1025*x[1]^4*x[2]^20 + x[2]^24
 
 julia> A, AtoR = affine_algebra(RG)
 (Quotient of multivariate polynomial ring by ideal (0), Hom: A -> graded multivariate polynomial ring)
diff --git a/test/book/specialized/decker-schmitt-invariant-theory/sym.jlcon b/test/book/specialized/decker-schmitt-invariant-theory/sym.jlcon
index 7377a2268415..cf1e7f8df41d 100644
--- a/test/book/specialized/decker-schmitt-invariant-theory/sym.jlcon
+++ b/test/book/specialized/decker-schmitt-invariant-theory/sym.jlcon
@@ -15,7 +15,7 @@ true
 julia> RS3 = invariant_ring(QQ, symmetric_group(3));
 
 julia> R = polynomial_ring(RS3)
-Multivariate polynomial ring in 3 variables over QQ graded by
+Multivariate polynomial ring in 3 variables over QQ graded by 
   x[1] -> [1]
   x[2] -> [1]
   x[3] -> [1]
diff --git a/test/book/specialized/eder-mohr-ideal-theoretic/gbeliminate.jlcon b/test/book/specialized/eder-mohr-ideal-theoretic/gbeliminate.jlcon
index 285858f608cd..c4027b70ecd6 100644
--- a/test/book/specialized/eder-mohr-ideal-theoretic/gbeliminate.jlcon
+++ b/test/book/specialized/eder-mohr-ideal-theoretic/gbeliminate.jlcon
@@ -6,12 +6,12 @@ julia> I = ideal(R, [x^2 + y + z - 1, x + y^2 + z - 1, x + y + z^2 - 1]);
 
 julia> groebner_basis(I, ordering = o)
 Gröbner basis with elements
-  1 -> z^6 - 4*z^4 + 4*z^3 - z^2
-  2 -> 2*y*z^2 + z^4 - z^2
-  3 -> x + y + z^2 - 1
-  4 -> y^2 + x + z - 1
+1 -> z^6 - 4*z^4 + 4*z^3 - z^2
+2 -> 2*y*z^2 + z^4 - z^2
+3 -> x + y + z^2 - 1
+4 -> y^2 + x + z - 1
 with respect to the ordering
-  degrevlex([x, y])*degrevlex([z])
+degrevlex([x, y])*degrevlex([z])
 
 julia> eliminate(I, [x,y]) # internally computes the above Gröbner basis
 Ideal generated by
@@ -19,8 +19,8 @@ Ideal generated by
 
 julia> groebner_basis(I)
 Gröbner basis with elements
-  1 -> z^2 + x + y - 1
-  2 -> y^2 + x + z - 1
-  3 -> x^2 + y + z - 1
+1 -> z^2 + x + y - 1
+2 -> y^2 + x + z - 1
+3 -> x^2 + y + z - 1
 with respect to the ordering
-  degrevlex([x, y, z])
+degrevlex([x, y, z])
diff --git a/test/book/specialized/eder-mohr-ideal-theoretic/nonproper.jlcon b/test/book/specialized/eder-mohr-ideal-theoretic/nonproper.jlcon
index ca08d4aa827c..b1911fd8d4c6 100644
--- a/test/book/specialized/eder-mohr-ideal-theoretic/nonproper.jlcon
+++ b/test/book/specialized/eder-mohr-ideal-theoretic/nonproper.jlcon
@@ -19,7 +19,7 @@ Ideal generated by
   x^2 - x*w
 
 julia> S = base_ring(clo)
-Multivariate polynomial ring in 5 variables over QQ graded by
+Multivariate polynomial ring in 5 variables over QQ graded by 
   x -> [1]
   y -> [1]
   z -> [1]
diff --git a/test/book/specialized/fang-fourier-monomial-bases/gap.jlcon b/test/book/specialized/fang-fourier-monomial-bases/gap.jlcon
new file mode 100644
index 000000000000..d47d03cf8252
--- /dev/null
+++ b/test/book/specialized/fang-fourier-monomial-bases/gap.jlcon
@@ -0,0 +1,2 @@
+L:= SimpleLieAlgebra("A", 4, Rationals);; 
+V:= HighestWeightModule(L, [1,3,2,1]);; 
diff --git a/test/book/specialized/fang-fourier-monomial-bases/oscar-vs-gap.jlcon b/test/book/specialized/fang-fourier-monomial-bases/oscar-vs-gap.jlcon
new file mode 100644
index 000000000000..cdf89abac0c2
--- /dev/null
+++ b/test/book/specialized/fang-fourier-monomial-bases/oscar-vs-gap.jlcon
@@ -0,0 +1 @@
+basis_lie_highest_weight(:A, 4, [1,3,2,1]);
diff --git a/test/book/specialized/fang-fourier-monomial-bases/sl7-cases.jlcon b/test/book/specialized/fang-fourier-monomial-bases/sl7-cases.jlcon
new file mode 100644
index 000000000000..d4c290bf899f
--- /dev/null
+++ b/test/book/specialized/fang-fourier-monomial-bases/sl7-cases.jlcon
@@ -0,0 +1,17 @@
+[fundamentals] => 320
+[fundamentals, [1, 0, 1, 0, 0]] => 70
+[fundamentals, [0, 0, 1, 0, 1]] => 70
+[fundamentals, [0, 1, 0, 1, 0]] => 54
+[fundamentals, [1, 0, 0, 1, 0]] => 28
+[fundamentals, [0, 1, 0, 0, 1]] => 28
+[fundamentals, [1, 0, 0, 1, 0], [0, 1, 0, 1, 0]] => 46
+[fundamentals, [0, 1, 0, 1, 0], [0, 1, 0, 0, 1]] => 46
+[fundamentals, [1, 0, 1, 0, 0], [1, 0, 0, 1, 0]] => 30
+[fundamentals, [0, 1, 0, 0, 1], [0, 0, 1, 0, 1]] => 30
+[fundamentals, [1, 0, 1, 0, 0], [0, 1, 0, 0, 1]] => 8
+[fundamentals, [1, 0, 0, 1, 0], [0, 0, 1, 0, 1]] => 8
+[fundamentals, [1, 0, 1, 0, 0], [1, 0, 0, 1, 0], [0, 1, 0, 1, 0]] => 70
+[fundamentals, [0, 1, 0, 1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 1]] => 70
+[fundamentals, [1, 0, 0, 1, 0], [0, 1, 0, 1, 0], [0, 1, 0, 0, 1]] => 10
+[fundamentals, [1, 0, 1, 0, 0], [1, 0, 0, 1, 0], [0, 1, 0, 1, 0], [0, 1, 0, 0, 1]] => 10
+[fundamentals, [1, 0, 0, 1, 0], [0, 1, 0, 1, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 1]] => 10
diff --git a/test/book/specialized/holt-ren-tropical-geometry/groebner.jlcon b/test/book/specialized/holt-ren-tropical-geometry/groebner.jlcon
index f414554cd3ee..866f45c64dc3 100644
--- a/test/book/specialized/holt-ren-tropical-geometry/groebner.jlcon
+++ b/test/book/specialized/holt-ren-tropical-geometry/groebner.jlcon
@@ -6,7 +6,7 @@ julia> nu_2 = tropical_semiring_map(QQ,2);
 
 julia> GBI = groebner_basis(I,nu_2,[0,0,0,0])
 2-element Vector{QQMPolyRingElem}:
- 3*x1 - x3 - 10*x4
+ x1 + 2*x2 - 3*x3
  3*x2 - 4*x3 + 5*x4
 
 julia> inI = initial(I,nu_2,[0,0,0,0])
@@ -17,7 +17,7 @@ Ideal generated by
 julia> GBI = groebner_basis(I,nu_2,[1,0,0,1])
 2-element Vector{QQMPolyRingElem}:
  3*x2 - 4*x3 + 5*x4
- -3*x1 + x3 + 10*x4
+ -x1 - 2*x2 + 3*x3
 
 julia> inI = initial(I,nu_2,[1,0,0,1])
 Ideal generated by
diff --git a/test/book/specialized/holt-ren-tropical-geometry/linearSpace1.jlcon b/test/book/specialized/holt-ren-tropical-geometry/linearSpace1.jlcon
index c09fec412ddd..89ecbed8d31e 100644
--- a/test/book/specialized/holt-ren-tropical-geometry/linearSpace1.jlcon
+++ b/test/book/specialized/holt-ren-tropical-geometry/linearSpace1.jlcon
@@ -9,15 +9,15 @@ Min tropical linear space
 
 julia> vertices_and_rays(TropL)
 5-element SubObjectIterator{Union{PointVector{QQFieldElem}, RayVector{QQFieldElem}}}:
- [0, 1, 0, 0]
+ [0, 0, 0, 0]
  [0, -1, -1, -1]
+ [0, 1, 0, 0]
  [0, 0, 1, 0]
  [0, 0, 0, 1]
- [0, 0, 0, 0]
 
 julia> maximal_polyhedra(IncidenceMatrix,TropL)
 4×5 IncidenceMatrix
+[1, 2]
+[1, 3]
+[1, 4]
 [1, 5]
-[2, 5]
-[3, 5]
-[4, 5]
diff --git a/test/book/specialized/joswig-kastner-lorenz-confirmable-workflows/polynomial-load.jlcon b/test/book/specialized/joswig-kastner-lorenz-confirmable-workflows/polynomial-load.jlcon
index 001284a2e2d1..cfe2de4da68f 100644
--- a/test/book/specialized/joswig-kastner-lorenz-confirmable-workflows/polynomial-load.jlcon
+++ b/test/book/specialized/joswig-kastner-lorenz-confirmable-workflows/polynomial-load.jlcon
@@ -2,7 +2,7 @@ julia> F, o = finite_field(7,2)
 (Finite field of degree 2 and characteristic 7, o)
 
 julia> R, (y,z) = F["y","z"]
-(Multivariate polynomial ring in 2 variables over F, FqMPolyRingElem[y, z])
+(Multivariate polynomial ring in 2 variables over GF(7, 2), FqMPolyRingElem[y, z])
 
 julia> save("p.mrdi", 2*y^3*z^4 + (o + 3)*z^2 + 5*o*y + 1)
 
diff --git a/test/book/specialized/kuehne-schroeter-matroids/ChowRings.jlcon b/test/book/specialized/kuehne-schroeter-matroids/ChowRings.jlcon
index 048441a4fb83..c2e3fe5d25c3 100644
--- a/test/book/specialized/kuehne-schroeter-matroids/ChowRings.jlcon
+++ b/test/book/specialized/kuehne-schroeter-matroids/ChowRings.jlcon
@@ -63,26 +63,26 @@ julia> RR, _ = graded_polynomial_ring(QQ,"y_#" => 1:length(basis_PD1));
 
 julia> map = hom(RR,AA,basis_PD1);
 
-julia> K = kernel(hom(RR,AA,[b^(rank(M)-2k)*b for b in basis_PD1]));
+julia> K = kernel(hom(RR,AA,[b^(rank(M)-2k)*b3 for b3 in basis_PD1]));
 
 julia> basis_HR = [map(h) for h in gens(K) if degree(h).coeff==k*g.coeff]
 7-element Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
- -x_{Edge(4, 3)} + 2*x_{Edge(2, 1),Edge(3, 1),Edge(3, 2)}
- -x_{Edge(4, 3)} + 2*x_{Edge(2, 1),Edge(4, 1),Edge(4, 2)}
- -x_{Edge(4, 3)} + 2*x_{Edge(2, 1),Edge(4, 3)}
- -x_{Edge(4, 3)} + 2*x_{Edge(3, 1),Edge(4, 2)}
- -x_{Edge(4, 3)} + 2*x_{Edge(3, 1),Edge(4, 1),Edge(4, 3)}
- -x_{Edge(4, 3)} + 2*x_{Edge(3, 2),Edge(4, 1)}
- -x_{Edge(4, 3)} + 2*x_{Edge(3, 2),Edge(4, 2),Edge(4, 3)}
+ x_{Edge(4, 3)}
+ -x_{Edge(2, 1),Edge(3, 1),Edge(3, 2)} + x_{Edge(2, 1),Edge(4, 1),Edge(4, 2)}
+ -x_{Edge(2, 1),Edge(3, 1),Edge(3, 2)} + 2*x_{Edge(2, 1),Edge(4, 3)}
+ -x_{Edge(2, 1),Edge(3, 1),Edge(3, 2)} + 2*x_{Edge(3, 1),Edge(4, 2)}
+ -x_{Edge(2, 1),Edge(3, 1),Edge(3, 2)} + x_{Edge(3, 1),Edge(4, 1),Edge(4, 3)}
+ -x_{Edge(2, 1),Edge(3, 1),Edge(3, 2)} + 2*x_{Edge(3, 2),Edge(4, 1)}
+ -x_{Edge(2, 1),Edge(3, 1),Edge(3, 2)} + x_{Edge(3, 2),Edge(4, 2),Edge(4, 3)}
 
 julia> Mat3 = matrix(QQ,[[(-1)^k*vol_map(b1*b^(rank(M)-2k-1)*b2) for b1 in basis_HR] for b2 in basis_HR])
-[6   2    4   2    4   2    4]
-[2   6    4   2    4   2    4]
-[4   4   10   4    6   4    6]
-[2   2    4   6    4   2    4]
-[4   4    6   4   10   4    6]
-[2   2    4   2    4   6    4]
-[4   4    6   4    6   4   10]
+[ 2   0   -2   0   -1   0   -1]
+[ 0   2    1   1    1   1    1]
+[-2   1    5   1    1   1    1]
+[ 0   1    1   5    1   1    1]
+[-1   1    1   1    2   1    1]
+[ 0   1    1   1    1   5    1]
+[-1   1    1   1    1   1    2]
 
 julia> is_positive_definite(matrix(ZZ,[ZZ(i) for i in Mat3]))
 true
diff --git a/test/book/specialized/kuehne-schroeter-matroids/realization_space.jlcon b/test/book/specialized/kuehne-schroeter-matroids/realization_space.jlcon
index 5aff6d593ef4..b6b82ceba9fa 100644
--- a/test/book/specialized/kuehne-schroeter-matroids/realization_space.jlcon
+++ b/test/book/specialized/kuehne-schroeter-matroids/realization_space.jlcon
@@ -124,5 +124,3 @@ One realization is given by
   [0   1   1   0   1   1   1   1   1]
   [0   0   0   1   2   3   6   3   2]
 in the rational field
-
-julia> results = collection = db = 0; # allow connection cleanup
diff --git a/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/draw_hypersurface.jlcon b/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/draw_hypersurface.jlcon
index 2bcf95ed6cc7..4847f599b04a 100644
--- a/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/draw_hypersurface.jlcon
+++ b/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/draw_hypersurface.jlcon
@@ -2,7 +2,7 @@ julia> Kt,t = rational_function_field(QQ,"t")
 (Rational function field over QQ, t)
 
 julia> Kxy, (x,y) = Kt["x", "y"]
-(Multivariate polynomial ring in 2 variables over Kt, AbstractAlgebra.Generic.MPoly{AbstractAlgebra.Generic.RationalFunctionFieldElem{QQFieldElem, QQPolyRingElem}}[x, y])
+(Multivariate polynomial ring in 2 variables over rational function field, AbstractAlgebra.Generic.MPoly{AbstractAlgebra.Generic.RationalFunctionFieldElem{QQFieldElem, QQPolyRingElem}}[x, y])
 
 julia> f=-x^3-4*x^2+y^2+(-8*t^4)*x
 -x^3 - 4*x^2 - 8*t^4*x + y^2
diff --git a/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/eliminate_xz.jlcon b/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/eliminate_xz.jlcon
index ca90d2698873..536e974116ef 100644
--- a/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/eliminate_xz.jlcon
+++ b/test/book/specialized/markwig-ristau-schleis-faithful-tropicalization/eliminate_xz.jlcon
@@ -2,7 +2,7 @@ julia> S,(b2,b34,b4,b56,b6,b7)=polynomial_ring(QQ,["b2", "b34", "b4", "b56", "b6
 (Multivariate polynomial ring in 6 variables over QQ, QQMPolyRingElem[b2, b34, b4, b56, b6, b7])
 
 julia> R,(x, y, z)=polynomial_ring(S,["x", "y", "z"])
-(Multivariate polynomial ring in 3 variables over S, AbstractAlgebra.Generic.MPoly{QQMPolyRingElem}[x, y, z])
+(Multivariate polynomial ring in 3 variables over multivariate polynomial ring, AbstractAlgebra.Generic.MPoly{QQMPolyRingElem}[x, y, z])
 
 julia> g = y^2-x*(x-b2^2)*(x-(b34+b4)^2)*(x-(b4)^2)*(x-(b6+b56)^2)*(x-(b6)^2)*(x+(b7)^2);
 
@@ -12,32 +12,4 @@ julia> Igf = ideal(R,[g,f]);
 
 julia> eliminate(Igf,[y])
 Ideal generated by
-  -x^7 + (b2^2 + b34^2 + 2*b34*b4 + 2*b4^2 + b56^2 + 2*b56*b6 + 2*b6^2)*x^6 + (-b2^2*b34^2 - 2*b2^2*b34*b4 - 2*b2^2*b4^2 - b2^2*b5
-  6^2 - 2*b2^2*b56*b6 - 2*b2^2*b6^2 + b2^2*b7^2 - b34^2*b4^2 - b34^2*b56^2 - 2*b34^2*b56*b6 - 2*b34^2*b6^2 + b34^2*b7^2 - 2*b34*b4
-  ^3 - 2*b34*b4*b56^2 - 4*b34*b4*b56*b6 - 4*b34*b4*b6^2 + 2*b34*b4*b7^2 - b4^4 - 2*b4^2*b56^2 - 4*b4^2*b56*b6 - 4*b4^2*b6^2 + 2*b4
-  ^2*b7^2 - b56^2*b6^2 + b56^2*b7^2 - 2*b56*b6^3 - b6^4)*x^5 + (b2^2*b34^2*b4^2 + b2^2*b34^2*b56^2 + 2*b2^2*b34^2*b56*b6 + 2*b2^2*
-  b34^2*b6^2 - b2^2*b34^2*b7^2 + 2*b2^2*b34*b4^3 + 2*b2^2*b34*b4*b56^2 + 4*b2^2*b34*b4*b56*b6 + 4*b2^2*b34*b4*b6^2 - 2*b2^2*b34*b4
-  *b7^2 + b2^2*b4^4 + 2*b2^2*b4^2*b56^2 + 4*b2^2*b4^2*b56*b6 + 4*b2^2*b4^2*b6^2 - 2*b2^2*b4^2*b7^2 + b2^2*b56^2*b6^2 - b2^2*b56^2*
-  b7^2 + 2*b2^2*b56*b6^3 - 2*b2^2*b56*b6*b7^2 + b2^2*b6^4 - 2*b2^2*b6^2*b7^2 + b34^2*b4^2*b56^2 + 2*b34^2*b4^2*b56*b6 + 2*b34^2*b4
-  ^2*b6^2 - b34^2*b4^2*b7^2 + b34^2*b56^2*b6^2 - b34^2*b56^2*b7^2 + 2*b34^2*b56*b6^3 - 2*b34^2*b56*b6*b7^2 + b34^2*b6^4 - 2*b34^2*
-  b6^2*b7^2 + 2*b34*b4^3*b56^2 + 4*b34*b4^3*b56*b6 + 4*b34*b4^3*b6^2 - 2*b34*b4^3*b7^2 + 2*b34*b4*b56^2*b6^2 - 2*b34*b4*b56^2*b7^2
-   + 4*b34*b4*b56*b6^3 - 2*b34*b4*b56*b6*b7^2 + 2*b34*b4*b6^4 - 2*b34*b4*b6^2*b7^2 + b4^4*b56^2 + 2*b4^4*b56*b6 + 2*b4^4*b6^2 - b4
-  ^4*b7^2 + 2*b4^2*b56^2*b6^2 - 2*b4^2*b56^2*b7^2 + 4*b4^2*b56*b6^3 - 2*b4^2*b56*b6*b7^2 + 2*b4^2*b6^4 - 2*b4^2*b6^2*b7^2)*x^4 + (
-  -b2^2*b34^2*b4^2*b56^2 - 2*b2^2*b34^2*b4^2*b56*b6 - 2*b2^2*b34^2*b4^2*b6^2 + b2^2*b34^2*b4^2*b7^2 - b2^2*b34^2*b56^2*b6^2 + b2^2
-  *b34^2*b56^2*b7^2 - 2*b2^2*b34^2*b56*b6^3 + 2*b2^2*b34^2*b56*b6*b7^2 - b2^2*b34^2*b6^4 + 2*b2^2*b34^2*b6^2*b7^2 - 2*b2^2*b34*b4^
-  3*b56^2 - 4*b2^2*b34*b4^3*b56*b6 - 4*b2^2*b34*b4^3*b6^2 + 2*b2^2*b34*b4^3*b7^2 - 2*b2^2*b34*b4*b56^2*b6^2 + 2*b2^2*b34*b4*b56^2*
-  b7^2 - 4*b2^2*b34*b4*b56*b6^3 + 4*b2^2*b34*b4*b56*b6*b7^2 - 2*b2^2*b34*b4*b6^4 + 4*b2^2*b34*b4*b6^2*b7^2 - b2^2*b4^4*b56^2 - 2*b
-  2^2*b4^4*b56*b6 - 2*b2^2*b4^4*b6^2 + b2^2*b4^4*b7^2 - 2*b2^2*b4^2*b56^2*b6^2 + 2*b2^2*b4^2*b56^2*b7^2 - 4*b2^2*b4^2*b56*b6^3 + 4
-  *b2^2*b4^2*b56*b6*b7^2 - 2*b2^2*b4^2*b6^4 + 4*b2^2*b4^2*b6^2*b7^2 + b2^2*b56^2*b6^2*b7^2 + 2*b2^2*b56*b6^3*b7^2 + b2^2*b6^4*b7^2
-   - b34^2*b4^2*b56^2*b6^2 + b34^2*b4^2*b56^2*b7^2 - 2*b34^2*b4^2*b56*b6^3 + 2*b34^2*b4^2*b56*b6*b7^2 - b34^2*b4^2*b6^4 + 2*b34^2*
-  b4^2*b6^2*b7^2 + b34^2*b56^2*b6^2*b7^2 + 2*b34^2*b56*b6^3*b7^2 + b34^2*b6^4*b7^2 - 2*b34*b4^3*b56^2*b6^2 + 2*b34*b4^3*b56^2*b7^2
-   - 4*b34*b4^3*b56*b6^3 + 4*b34*b4^3*b56*b6*b7^2 - 2*b34*b4^3*b6^4 + 4*b34*b4^3*b6^2*b7^2 - b4^4*b56^2*b6^2 + b4^4*b56^2*b7^2 - 2
-  *b4^4*b56*b6^3 + 2*b4^4*b56*b6*b7^2 - b4^4*b6^4 + 2*b4^4*b6^2*b7^2)*x^3 + (b2^2*b34^2*b4^2*b56^2*b6^2 - b2^2*b34^2*b4^2*b56^2*b7
-  ^2 + 2*b2^2*b34^2*b4^2*b56*b6^3 - 2*b2^2*b34^2*b4^2*b56*b6*b7^2 + b2^2*b34^2*b4^2*b6^4 - 2*b2^2*b34^2*b4^2*b6^2*b7^2 - b2^2*b34^
-  2*b56^2*b6^2*b7^2 - 2*b2^2*b34^2*b56*b6^3*b7^2 - b2^2*b34^2*b6^4*b7^2 + 2*b2^2*b34*b4^3*b56^2*b6^2 - 2*b2^2*b34*b4^3*b56^2*b7^2 
-  + 4*b2^2*b34*b4^3*b56*b6^3 - 4*b2^2*b34*b4^3*b56*b6*b7^2 + 2*b2^2*b34*b4^3*b6^4 - 4*b2^2*b34*b4^3*b6^2*b7^2 - 2*b2^2*b34*b4*b56^
-  2*b6^2*b7^2 - 4*b2^2*b34*b4*b56*b6^3*b7^2 - 2*b2^2*b34*b4*b6^4*b7^2 + b2^2*b4^4*b56^2*b6^2 - b2^2*b4^4*b56^2*b7^2 + 2*b2^2*b4^4*
-  b56*b6^3 - 2*b2^2*b4^4*b56*b6*b7^2 + b2^2*b4^4*b6^4 - 2*b2^2*b4^4*b6^2*b7^2 - 2*b2^2*b4^2*b56^2*b6^2*b7^2 - 4*b2^2*b4^2*b56*b6^3
-  *b7^2 - 2*b2^2*b4^2*b6^4*b7^2)*x^2 + (b2^2*b34^2*b4^2*b56^2*b6^2*b7^2 + 2*b2^2*b34^2*b4^2*b56*b6^3*b7^2 + b2^2*b34^2*b4^2*b6^4*b
-  7^2 + 2*b2^2*b34*b4^3*b56^2*b6^2*b7^2 + 4*b2^2*b34*b4^3*b56*b6^3*b7^2 + 2*b2^2*b34*b4^3*b6^4*b7^2 + b2^2*b4^4*b56^2*b6^2*b7^2 + 
-  2*b2^2*b4^4*b56*b6^3*b7^2 + b2^2*b4^4*b6^4*b7^2)*x
+  -x^7 + (b2^2 + b34^2 + 2*b34*b4 + 2*b4^2 + b56^2 + 2*b56*b6 + 2*b6^2)*x^6 + (-b2^2*b34^2 - 2*b2^2*b34*b4 - 2*b2^2*b4^2 - b2^2*b56^2 - 2*b2^2*b56*b6 - 2*b2^2*b6^2 + b2^2*b7^2 - b34^2*b4^2 - b34^2*b56^2 - 2*b34^2*b56*b6 - 2*b34^2*b6^2 + b34^2*b7^2 - 2*b34*b4^3 - 2*b34*b4*b56^2 - 4*b34*b4*b56*b6 - 4*b34*b4*b6^2 + 2*b34*b4*b7^2 - b4^4 - 2*b4^2*b56^2 - 4*b4^2*b56*b6 - 4*b4^2*b6^2 + 2*b4^2*b7^2 - b56^2*b6^2 + b56^2*b7^2 - 2*b56*b6^3 - b6^4)*x^5 + (b2^2*b34^2*b4^2 + b2^2*b34^2*b56^2 + 2*b2^2*b34^2*b56*b6 + 2*b2^2*b34^2*b6^2 - b2^2*b34^2*b7^2 + 2*b2^2*b34*b4^3 + 2*b2^2*b34*b4*b56^2 + 4*b2^2*b34*b4*b56*b6 + 4*b2^2*b34*b4*b6^2 - 2*b2^2*b34*b4*b7^2 + b2^2*b4^4 + 2*b2^2*b4^2*b56^2 + 4*b2^2*b4^2*b56*b6 + 4*b2^2*b4^2*b6^2 - 2*b2^2*b4^2*b7^2 + b2^2*b56^2*b6^2 - b2^2*b56^2*b7^2 + 2*b2^2*b56*b6^3 - 2*b2^2*b56*b6*b7^2 + b2^2*b6^4 - 2*b2^2*b6^2*b7^2 + b34^2*b4^2*b56^2 + 2*b34^2*b4^2*b56*b6 + 2*b34^2*b4^2*b6^2 - b34^2*b4^2*b7^2 + b34^2*b56^2*b6^2 - b34^2*b56^2*b7^2 + 2*b34^2*b56*b6^3 - 2*b34^2*b56*b6*b7^2 + b34^2*b6^4 - 2*b34^2*b6^2*b7^2 + 2*b34*b4^3*b56^2 + 4*b34*b4^3*b56*b6 + 4*b34*b4^3*b6^2 - 2*b34*b4^3*b7^2 + 2*b34*b4*b56^2*b6^2 - 2*b34*b4*b56^2*b7^2 + 4*b34*b4*b56*b6^3 - 2*b34*b4*b56*b6*b7^2 + 2*b34*b4*b6^4 - 2*b34*b4*b6^2*b7^2 + b4^4*b56^2 + 2*b4^4*b56*b6 + 2*b4^4*b6^2 - b4^4*b7^2 + 2*b4^2*b56^2*b6^2 - 2*b4^2*b56^2*b7^2 + 4*b4^2*b56*b6^3 - 2*b4^2*b56*b6*b7^2 + 2*b4^2*b6^4 - 2*b4^2*b6^2*b7^2)*x^4 + (-b2^2*b34^2*b4^2*b56^2 - 2*b2^2*b34^2*b4^2*b56*b6 - 2*b2^2*b34^2*b4^2*b6^2 + b2^2*b34^2*b4^2*b7^2 - b2^2*b34^2*b56^2*b6^2 + b2^2*b34^2*b56^2*b7^2 - 2*b2^2*b34^2*b56*b6^3 + 2*b2^2*b34^2*b56*b6*b7^2 - b2^2*b34^2*b6^4 + 2*b2^2*b34^2*b6^2*b7^2 - 2*b2^2*b34*b4^3*b56^2 - 4*b2^2*b34*b4^3*b56*b6 - 4*b2^2*b34*b4^3*b6^2 + 2*b2^2*b34*b4^3*b7^2 - 2*b2^2*b34*b4*b56^2*b6^2 + 2*b2^2*b34*b4*b56^2*b7^2 - 4*b2^2*b34*b4*b56*b6^3 + 4*b2^2*b34*b4*b56*b6*b7^2 - 2*b2^2*b34*b4*b6^4 + 4*b2^2*b34*b4*b6^2*b7^2 - b2^2*b4^4*b56^2 - 2*b2^2*b4^4*b56*b6 - 2*b2^2*b4^4*b6^2 + b2^2*b4^4*b7^2 - 2*b2^2*b4^2*b56^2*b6^2 + 2*b2^2*b4^2*b56^2*b7^2 - 4*b2^2*b4^2*b56*b6^3 + 4*b2^2*b4^2*b56*b6*b7^2 - 2*b2^2*b4^2*b6^4 + 4*b2^2*b4^2*b6^2*b7^2 + b2^2*b56^2*b6^2*b7^2 + 2*b2^2*b56*b6^3*b7^2 + b2^2*b6^4*b7^2 - b34^2*b4^2*b56^2*b6^2 + b34^2*b4^2*b56^2*b7^2 - 2*b34^2*b4^2*b56*b6^3 + 2*b34^2*b4^2*b56*b6*b7^2 - b34^2*b4^2*b6^4 + 2*b34^2*b4^2*b6^2*b7^2 + b34^2*b56^2*b6^2*b7^2 + 2*b34^2*b56*b6^3*b7^2 + b34^2*b6^4*b7^2 - 2*b34*b4^3*b56^2*b6^2 + 2*b34*b4^3*b56^2*b7^2 - 4*b34*b4^3*b56*b6^3 + 4*b34*b4^3*b56*b6*b7^2 - 2*b34*b4^3*b6^4 + 4*b34*b4^3*b6^2*b7^2 - b4^4*b56^2*b6^2 + b4^4*b56^2*b7^2 - 2*b4^4*b56*b6^3 + 2*b4^4*b56*b6*b7^2 - b4^4*b6^4 + 2*b4^4*b6^2*b7^2)*x^3 + (b2^2*b34^2*b4^2*b56^2*b6^2 - b2^2*b34^2*b4^2*b56^2*b7^2 + 2*b2^2*b34^2*b4^2*b56*b6^3 - 2*b2^2*b34^2*b4^2*b56*b6*b7^2 + b2^2*b34^2*b4^2*b6^4 - 2*b2^2*b34^2*b4^2*b6^2*b7^2 - b2^2*b34^2*b56^2*b6^2*b7^2 - 2*b2^2*b34^2*b56*b6^3*b7^2 - b2^2*b34^2*b6^4*b7^2 + 2*b2^2*b34*b4^3*b56^2*b6^2 - 2*b2^2*b34*b4^3*b56^2*b7^2 + 4*b2^2*b34*b4^3*b56*b6^3 - 4*b2^2*b34*b4^3*b56*b6*b7^2 + 2*b2^2*b34*b4^3*b6^4 - 4*b2^2*b34*b4^3*b6^2*b7^2 - 2*b2^2*b34*b4*b56^2*b6^2*b7^2 - 4*b2^2*b34*b4*b56*b6^3*b7^2 - 2*b2^2*b34*b4*b6^4*b7^2 + b2^2*b4^4*b56^2*b6^2 - b2^2*b4^4*b56^2*b7^2 + 2*b2^2*b4^4*b56*b6^3 - 2*b2^2*b4^4*b56*b6*b7^2 + b2^2*b4^4*b6^4 - 2*b2^2*b4^4*b6^2*b7^2 - 2*b2^2*b4^2*b56^2*b6^2*b7^2 - 4*b2^2*b4^2*b56*b6^3*b7^2 - 2*b2^2*b4^2*b6^4*b7^2)*x^2 + (b2^2*b34^2*b4^2*b56^2*b6^2*b7^2 + 2*b2^2*b34^2*b4^2*b56*b6^3*b7^2 + b2^2*b34^2*b4^2*b6^4*b7^2 + 2*b2^2*b34*b4^3*b56^2*b6^2*b7^2 + 4*b2^2*b34*b4^3*b56*b6^3*b7^2 + 2*b2^2*b34*b4^3*b6^4*b7^2 + b2^2*b4^4*b56^2*b6^2*b7^2 + 2*b2^2*b4^4*b56*b6^3*b7^2 + b2^2*b4^4*b6^4*b7^2)*x