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<!DOCTYPE html>
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Finite fields · Singular.jl</title><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">Singular.jl</a></span></div><form class="docs-search" action="../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../">Getting Started</a></li><li><span class="tocitem">Coefficient rings</span><ul><li><a class="tocitem" href="../integer/">Integers</a></li><li><a class="tocitem" href="../rational/">Rational field</a></li><li><a class="tocitem" href="../modn/">Integers mod n</a></li><li><a class="tocitem" href="../modp/">Integers mod p</a></li><li><a class="tocitem" href="../transExt/">Function fields</a></li><li class="is-active"><a class="tocitem" href>Finite fields</a><ul class="internal"><li><a class="tocitem" href="#Finite-field-functionality"><span>Finite field functionality</span></a></li></ul></li><li><a class="tocitem" href="../nemo/">Nemo rings and fields</a></li></ul></li><li><a class="tocitem" href="../polynomial/">Multivariate polynomials</a></li><li><a class="tocitem" href="../noncommutative/">Noncommutative algebras</a></li><li><a class="tocitem" href="../ideal/">Ideals</a></li><li><a class="tocitem" href="../qring/">Quotient Rings</a></li><li><span class="tocitem">Modules</span><ul><li><a class="tocitem" href="../module/">Finitely generated modules</a></li><li><a class="tocitem" href="../vector/">Free modules and vectors</a></li></ul></li><li><a class="tocitem" href="../alghom/">Algebra Homomorphisms</a></li><li><a class="tocitem" href="../resolution/">Resolutions</a></li><li><a class="tocitem" href="../matrix/">Matrices</a></li><li><a class="tocitem" href="../caller/">Interpreter Functionality</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Coefficient rings</a></li><li class="is-active"><a href>Finite fields</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Finite fields</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/oscar-system/Singular.jl/blob/master/docs/src/GF.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Finite-fields"><a class="docs-heading-anchor" href="#Finite-fields">Finite fields</a><a id="Finite-fields-1"></a><a class="docs-heading-anchor-permalink" href="#Finite-fields" title="Permalink"></a></h1><p>Finite fields are implemented via the Singular <code>n_GF</code> type for any characteristic and degree contained in the Singular Conway tables.</p><p>The associated finite field is represented by a parent object which can be constructed by a call to the <code>FiniteField</code> constructor.</p><p>The types of the parent objects and elements of the associated finite fields are given in the following table according to the library providing them.</p><table><tr><th style="text-align: right">Library</th><th style="text-align: right">Element type</th><th style="text-align: right">Parent type</th></tr><tr><td style="text-align: right">Singular</td><td style="text-align: right"><code>n_GF</code></td><td style="text-align: right"><code>Singular.N_GField</code></td></tr></table><p>All finite field element types belong directly to the abstract type <code>FieldElem</code> and all the parent object types belong to the abstract type <code>Field</code>.</p><h2 id="Finite-field-functionality"><a class="docs-heading-anchor" href="#Finite-field-functionality">Finite field functionality</a><a id="Finite-field-functionality-1"></a><a class="docs-heading-anchor-permalink" href="#Finite-field-functionality" title="Permalink"></a></h2><p>Singular.jl finite fields implement the Field interface of AbstractAlgebra.jl.</p><p><a href="https://nemocas.github.io/AbstractAlgebra.jl/fields.html">https://nemocas.github.io/AbstractAlgebra.jl/fields.html</a></p><p>Below, we describe the functionality that is specific to Singular finite field and not already listed at the given link.</p><h3 id="Constructors"><a class="docs-heading-anchor" href="#Constructors">Constructors</a><a id="Constructors-1"></a><a class="docs-heading-anchor-permalink" href="#Constructors" title="Permalink"></a></h3><p>The following constructors are available to create finite fields and their elements.</p><article class="docstring"><header><a class="docstring-binding" id="Singular.FiniteField-Tuple{Int64, Int64, Union{Char, AbstractString, Symbol}}" href="#Singular.FiniteField-Tuple{Int64, Int64, Union{Char, AbstractString, Symbol}}"><code>Singular.FiniteField</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">FiniteField(p::Int, n::Int, S::VarName; cached=true)</code></pre><p>Return a tuple <code>K, a</code> consisting of a finite field <code>K</code> of characteristic <span>$p$</span> and degree <span>$n$</span>, and its generator <code>a</code>. The string used to print the generator is given by <code>S</code>. If the finite field is not listed in the Conway tables included in Singular, an error will be raised. By default, finite fields are cached globally, so that there is only one finite field in the system with given characteristic, degree and string. If this is not the desired behaviour, one can pass <code>false</code> for the optional <code>cached</code> parameter.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/oscar-system/Singular.jl/blob/8eaafd6ebc21da317e7e97466308e8acda388d9f/src/number/n_GF.jl#L334-L344">source</a></section></article><p>Given a finite field <span>$R$</span>, we also have the following coercions in addition to the standard ones expected.</p><pre><code class="language-julia hljs">R(n::ZZRingElem)</code></pre><p>Coerce a Flint integer value into the field.</p><h3 id="Basic-manipulation"><a class="docs-heading-anchor" href="#Basic-manipulation">Basic manipulation</a><a id="Basic-manipulation-1"></a><a class="docs-heading-anchor-permalink" href="#Basic-manipulation" title="Permalink"></a></h3><article class="docstring"><header><a class="docstring-binding" id="AbstractAlgebra.degree-Tuple{N_GField}" href="#AbstractAlgebra.degree-Tuple{N_GField}"><code>AbstractAlgebra.degree</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">degree(R::N_GField)</code></pre><p>Return the degree of the field as an extension of <span>$\mathbb{F}_p$</span>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/oscar-system/Singular.jl/blob/8eaafd6ebc21da317e7e97466308e8acda388d9f/src/number/n_GF.jl#L30-L34">source</a></section></article><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; R,w = FiniteField(7, 2, &quot;w&quot;)
(Finite Field of Characteristic 7 and degree 2, w)

julia&gt; w^48 == 1
true

julia&gt; a = R(5)
w^40

julia&gt; is_unit(a)
true

julia&gt; c = characteristic(R)
7

julia&gt; d = degree(R)
2</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../transExt/">« Function fields</a><a class="docs-footer-nextpage" href="../nemo/">Nemo rings and fields »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Friday 25 August 2023 19:03">Friday 25 August 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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