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field.tex
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field.tex
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\chapter{Field}
\section{Field Extension}
\begin{definition}{Field}{}
A \textbf{field} is a commutative ring $K$ such that $K^{\times}=K-\{0\}$.
\end{definition}
\begin{proposition}{}{}
If $K$ is a field, $R$a ring homomorphism $f:K\to R$.
\end{proposition}
\begin{definition}{Field Extension}{}
Let $K$ be a field and $L$ be a commutative ring.
\end{definition}
\begin{lemma}{Zariski's Lemma}{}
If a field is a finite-type $K$-algebra, then it is a finite extension of $K$ (that is, it is also finitely generated as a $K$-linear space).
\end{lemma}
\begin{theorem}{Hilbert's Weak Nullstellensatz}{}
If $\overline{\Bbbk}$ is an algebraically closed field, then the maximal ideals of $\overline{\Bbbk}\left[x_1, \cdots, x_n\right]$ are precisely those ideals of the form $\left(x_1-a_1, \cdots, x_n-a_{n}\right)$, where $a_i\in \overline{\Bbbk}$.
\end{theorem}
\begin{theorem}{Hilbert's Nullstellensatz}{}
If $\Bbbk$ is any field and $\mathfrak{m}$ is a maximal ideal of $\Bbbk\left[x_1, \ldots, x_n\right]$, then $\Bbbk\left[x_1, \ldots, x_n\right]/\mathfrak{m}$ is a finite extension of $\Bbbk$.
\end{theorem}