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\begin{document}
\title{MATH 172, Fall 2017, Final Exam - Take home}

\maketitle


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\begin{center}
{\bf Please print your name clearly!}
\end{center}
\bigskip

\noindent
{\bf Name:\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }}
\bigskip

\noindent
{\it This test is due on Wednesday, 12/6/17, in class. While completing it, feel free to use your lecture notes from class, as well as your textbook. You are however not allowed to consult with anyone: it is understood that solutions to this exam represent solely your own work with no outside assistance. Good luck!}
\bigskip

\noindent
{\bf Problem 1 	(15 points).} Let $F$ be a perfect field, and let $p(x) \in F[x]$ be an irreducible polynomial of degree 37. Suppose that $K := F[x]/\left< p(x) \right>$ is the splitting field of $p(x)$. Prove that for every subfield $B$ of $K$ which contains $F$, the extension $B/F$ is Galois.
\bigskip

\newpage

\noindent
{\bf Problem 2 (25 points).} Let $f(x) = x^3-7$.
\smallskip

\noindent
{\bf Part a - (15 points).} Let $K$ be the splitting field of $f(x)$ over $\que$. Describe the Galois group of $K/\que$, and describe all the intermediate fields $E$, i.e. subfields of $K$ containing $\que$. Which of them are Galois and which are not?
\smallskip

\noindent
{\bf Part b - (10 points).} Let $L$ be the splitting field of $f(x)$ over $\real$. Describe the Galois group of $L/\real$.
\smallskip


\newpage

\noindent
{\bf Problem 3 (15 points).} 
\smallskip

\noindent
{\bf Part a - (5 points).} Describe all maximal ideals in $\cee[x]$.
\smallskip

\noindent
{\bf Part b - (5 points).} Describe all maximal ideals in $\real[x]$.
\smallskip

\noindent
{\bf Part c - (5 points).} Describe (up to isomorphism) the fields that may be obtained as quotients $\real[x]/M$, where $M$ is a maximal ideal in~$\real[x]$.

\newpage

\noindent
{\bf Problem 4 (5 points).} Let $f(x) \in \que[x]$ be a polynomial of degree 3, and $G$ its Galois group. Prove that if $G$ has order 3, then $f(x)$  splits completely over~$\real$.

\newpage


\noindent
{\bf Problem 5 (20 points).} Let $R$ be a PID and $F$ its field of fractions. Suppose $S$ is a ring such that $R \subset S \subset F$.
\smallskip

\noindent
{\bf Part a - (10 points).} Prove that all elements of $S$ can be written as $a/b$ with $a,b \in R$ and $1/b \in S$.
\smallskip

\noindent
{\bf Part b - (10 points).} Prove that $S$ is a PID.


\newpage


\noindent
{\bf Problem 6 (35 points).} Let $K$ be the splitting field of $x^4-x^2-1$ over~$\que$.
\smallskip

\noindent
{\bf Part a - (15 points).} Find the Galois group of $K/F$.
\smallskip

\noindent
{\bf Part a - (20 points).} Give example of two subfields $A$ and $B$ of $K$ so that $A/\que$ is Galois and $B/\que$ is not. Compute the Galois group of~$A/\que$.

\newpage

\noindent
{\bf Problem 7 (40 points).} {\it Remark: This problem is closely connected to a problem we have seen on the midterm, however goes in the reverse direction.}
\smallskip

A PID $R$ is called a {\it discrete valuation ring} (DVR) if it has precisely one non-zero prime ideal. Throughout this problem, let $R$ be a DVR, $P$ its unique prime ideal, and $K = \frck(R)$ its field of fractions.
\bigskip

\noindent
{\bf Part a - (10 points).} Show that for any non-zero element $x \in K$ either $x \in R$ or $x^{-1} \in R$.
\bigskip

\noindent
{\bf Part b - (10 points).} Show that $\bigcap_{n \geq 1} P^n = 0$, where
$$P^n = \left\{ a_1 a_2 \dots a_n : a_i \in P\ \forall\ 1 \leq i \leq n \right\}.$$
\smallskip

\noindent
{\bf Part c - (10 points).} Show that if $f(x) \in R[x]$ is a monic polynomial which has a root $a \in K$, then $a \in R$. Rings having this property are called {\it integrally closed}.
\bigskip

\noindent
{\bf Part d - (10 points).} Prove that $K$ cannot be algebraically closed.

\newpage

\noindent
{\bf Problem 8 (15 points).} Let $R$ be a UFD. 
\smallskip

\noindent
{\bf Part a - (5 points).} Prove that $R[x]$ is a UFD (you can use Gauss' Lemma and the fact that $F[x]$ is a PID when $F$ is a field without proving them).
\smallskip

\noindent
{\bf Part b - (10 points).} Suppose that for any $a, b \in R$ the ideal $\left< a,b \right>$ is principal. Prove that $R$ is a PID.

\end{document}