\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}
