Created the Abstract Algebra theorems and definitions cheat sheet

part-4/chapters/chapter-20/chapter-20.tex

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+\chapter{Extension Fields}
+\subimport{./}{the-fundamental-theorem-of-field-theory.tex}
+\subimport{./}{splitting-fields.tex}
+\subimport{./}{zeros-of-an-irreducible-polynomial.tex}

part-4/chapters/chapter-20/splitting-fields.tex

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+\section{Splitting Fields}
+
+\begin{definition}[Splitting Field]
+	Let $\E$ be an extension field of $\F$ and let $f(x) \in \F[x]$ with degree at least 1. We say that $f(x)$ \textit{splits} in $\E$ if there are elements $a \in \F$ and $a_1,a_2,\dots,a_n \in \E$ such that
+	\[ f(x) = a(x-a_1)(x-a_2)\dots(x-a_n) \]
+	We call $\E$ a \textit{splitting field for $f(x)$ over $\F$} if
+	\[ \E = \F(a_1,a_2,\dots,a_n) \]
+\end{definition}
+
+\begin{theorem}[Existence of Splitting Fields]
+	Let $\F$ be a field and let $f(x)$ be a nonconstant element of $\F[x]$. Then there exists a splitting field $\E$ for $f(x)$ over $\F$.
+\end{theorem}
+
+\begin{theorem}[$\mathbf{\F(a) \approx \F[x]/\lr{p(x)}}$]
+	Let $\F$ be a field and let $p(x) \in \F[x]$ be irreducible over $\F$. If $a$ is a zero of $p(x)$ in some extension $\E$ of $\F$, then $\F(a)$ is isomorphic to $\F[x]/\lr{p(x)}$. Furthermore, if $\deg p(x) = n$, then every member of $\F(a)$ can be uniquely expressed in the form
+	\[ c_{n-1}a^{n-1}+c_{n-2}a^{n-2}+\dots+c_1a+c_0 \]
+	where $c_0,c_1,\dots,c_{n-1} \in \F$.
+\end{theorem}
+
+\begin{corollary}[$\mathbf{\F(a) \approx \F(b)}$]
+	Let $\F$ be a field and let $p(x) \in \F[x]$ be irreducible over $\F$. If $a$ is a zero of $p(x)$ in some extension $\E$ of $\F$ and $b$ is a zero of $p(x)$ in some extension $\E'$ of $\F$, then the fields $\F(a)$ and $\F(b)$ are isomorphic.
+\end{corollary}
+
+\begin{lemma}
+	Let $\F$ be a field, let $p(x) \in \F[x]$ be irreducible over $\F$, and let $a$ be a zero of $p(x)$ in some extension of $\F$. If $\phi$ is a field isomorphism from $\F$ to $\F'$ and $b$ is a zero of $\phi(p(x))$ in some extension of $\F'$, then there is an isomorphism from $\F(a)$ to $\F'(b)$ that agrees with $\phi$ on $\F$ and carries $a$ to $b$.
+\end{lemma}
+
+\begin{theorem}[Extending $\mathbf{\phi: \F \to \F'}$]
+	Let $\phi$ be an isomorphism from a field $\F$ to a field $\F'$ and let $f(x) \in \F[x]$. If $\E$ is a splitting field for $f(x)$ over $\F$ and $\E'$ is a splitting field for $\phi(f(x))$ over $\F'$, then there is an isomorphism from $\E$ to $\E'$ that agrees with $\phi$ on $\F$.
+\end{theorem}
+
+\begin{corollary}[Splitting Fields Are Unique]
+	Let $\F$ be a field and let $f(x) \in \F[x]$. Then any two splitting fields of $f(x)$ over $\F$ are isomorphic.
+\end{corollary}

part-4/chapters/chapter-20/the-fundamental-theorem-of-field-theory.tex

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+\section{The Fundamental Theorem of Field Theory}
+
+\begin{definition}[Extension Field]
+	A field $\E$ is an \textit{extension field} of a field $\F$ if $\F \subseteq \E$ and the operations of $\F$ are those of $\E$ restricted to $\F$.
+\end{definition}
+
+\begin{theorem}[Fundamental Theorem of Field Theory (Kronecker's Theorem, 1887)]
+	Let $\F$ be a field and let $f(x)$ be a nonconstant polynomial in $\F[x]$. Then there is an extension field $\E$ of $\F$ in which $f(x)$ has a zero.
+\end{theorem}

part-4/chapters/chapter-20/zeros-of-an-irreducible-polynomial.tex

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+\section{Zeros of an Irreducible Polynomial}
+
+\begin{definition}[Derivative]
+	Let $f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$ belong to $\F[x]$. The \textit{derivative} of $f(x)$, denoted by $f'(x)$, is the polynomial $na_nx^{x-1} + (n-1)a_{n-1}x^{n-2} + \dots + a_1$ in $\F[x]$.
+\end{definition}
+
+\begin{lemma}[Properties of the Derivative]
+	Let $f(x)$ and $g(x) \in \F[x]$ and let $a \in \F$. Then
+	\begin{enumerate}
+		\item $(f(x) + g(x))' = f'(x) + g'(x)$.
+		\item $(af(x))' = af'(x)$.
+		\item $(f(x)g(x))' = f(x)g'(x) + g(x)f'(x)$.
+	\end{enumerate}
+\end{lemma}
+
+\begin{theorem}[Criterion for Multiple Zeros]
+	A polynomial $f(x)$ over a field $\F$ has a multiple zero in some extension $\E$ if and only if $f(x)$ and $f'(x)$ have a common factor of positive degree in $\F[x]$.
+\end{theorem}
+
+\begin{theorem}[Zeros of an Irreducible]
+	Let $f(x)$ be an irreducible polynomial over a field $\F$. If $\F$ has characteristic 0, then $f(x)$ has no multiple zeros. If $\F$ has characteristic $p \neq 0$, then $f(x)$ has a multiple zero if it is of the form $f(x) = g(x^p)$ for some $g(x)$ in $\F[x]$.
+\end{theorem}
+
+\begin{definition}[Perfect Field]
+	A field $\F$ is called \textit{perfect} if $\F$ has characteristic 0 or if $\F$ has characteristic $p$ and $\F^p=\{a^p\ \vert\ a \in \F\} = \F$.
+\end{definition}
+
+\begin{theorem}[Finite Fields Are Perfect]
+	Every finite field is perfect.
+\end{theorem}
+
+\begin{theorem}[Criterion for No Multiple Zeros]
+	If $f(x)$ is an irreducible polynomial over a perfect field $\F$, then $f(x)$ has no multiple zeros.
+\end{theorem}
+
+\begin{theorem}[Zeros of an Irreducible over a Splitting Field]
+	Let $f(x)$ be an irreducible polynomial over a field $\F$ and let $\E$ be a splitting field of $f(x)$ over $\F$. Then all the zeros of $f(x)$ in $\E$ have the same multiplicity.
+\end{theorem}
+
+\begin{corollary}[Factorization of an Irreducible over a Splitting Field]
+	Let $f(x)$ be an irreducible polynomial over a field $\F$ and let $\E$ be a splitting field of $f(x)$. Then $f(x)$ has the form
+	\[ a(x-a_1)^n(x-a_2)^n\dots(x-a_t)^n \]
+	where $a_1,a_2,\dots,a_t$ are distinct elements of $\E$ and $a \in \F$.
+\end{corollary}