\documentclass{article}
\usepackage{amsmath, amssymb,verbatim, amsthm, graphicx, mathtools}
\usepackage[all]{xy}

%exercise environment
\newcounter{exercise}
\setcounter{exercise}{0}
\newenvironment{exercise}{\refstepcounter{exercise} \noindent{\bf{Exercise \theexercise.}}}{\vspace{.5cm}}

\newtheorem*{thm*}{Theorem}

\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\ord}{\operatorname{ord}}
\newcommand{\vp}{\varphi}

\textwidth 6.5in
\textheight 9in
\oddsidemargin 0in
\evensidemargin 0in
\headsep -0.5in

\setlength{\parskip}{0.25cm}

\pagenumbering{gobble}

\begin{document}

\begin{center}
\includegraphics[width=0.5in]{../ltr-logo.jpg}
\end{center}

\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 10.2  \\ Fall 2019 \hfill Due November 20}}}

\large

\vspace{1cm}

\begin{exercise}
Let $F$ be a field.  

\begin{itemize}

\item[\bf a.]  If $a \in F$ and $a \ne \Box$, show that $X^2 - a \in F[X]$ is irreducible.

\item[\bf b.]  Let $a \in F$ with $a \ne \Box$, and let $\sqrt{a}$ denote a fixed root of $X^2 - a$ in some extension $K/F$.  Prove that 
\[
k=\left\{ x + y \sqrt{a} \, \Big| \, x, y \in F \right\}
\]
is a subfield of $K$.  Conclude that $\displaystyle k = F\left(\sqrt{a}\right)$.

\item[\bf c.]  Let $a, b \in F$ with $a, b \ne \Box$.  Let $K/F$ be an extension containing both $\sqrt{a}$ and $\sqrt{b}$.  Prove that $\displaystyle F \left(\sqrt{a} \right) = F\big(\sqrt{b}\big)$ if and only if there is a $c \in F^{\times}$ so that $a = b c^2$.

\end{itemize}

\end{exercise}

\begin{exercise}
Let $F$ be a field and let $f_1, \ldots, f_n \in F[X]$.  Prove that a field $K/F$ is a splitting field over $F$ for the set $\{f_1, f_2, \ldots, f_n \}$ if and only if $K$ is a splitting field over $F$ for the single polynomial $f_1 f_2 \cdots f_n$.
\end{exercise}

\begin{exercise}
Lek $K/k/F$ be fields and let $f \in F[X]$.  Prove that if $K$ is a splitting field of $f$ over $F$, then $K$ is a splitting field of $f$ over $k$.
\end{exercise}


\end{document}
