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\begin{document}
\title{MATH 195, Spring 2019, Take-home final}

\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 Tuesday, 5/7/19, in class. While completing it, feel free to use your lecture notes from class, as well as those posted on the class webpage. You are however not allowed to consult with anyone: it is understood that solutions to this midterm represent solely your own work with no outside assistance. Additionally, if you use a homework exercise, please include its solution. Good luck!}
\bigskip


\noindent
{\bf Problem 1. (40 points)} Consider the number field $K = \que(\sqrt{3},\sqrt{-3})$.
\medskip

\noindent
{\bf Part a (10 points)} Determine the degree of $K$ over $\que$. Prove your answer.
\medskip

\noindent
{\bf Part b (10 points)} Find a primitive element for $K$ over $\que$. Prove your answer.
\medskip

\noindent
{\bf Part c (10 points)} Find the minimal polynomial of the primitive element you found in part b. Prove your answer.
\medskip

\noindent
{\bf Part d (10 points)} Is $2$ a prime in~$K$? Prove your answer.
\bigskip




\newpage



\noindent
{\bf Problem 2. (10 points)} Let $K=\que(\sqrt{-13})$. Is $\O_K$ a PID (principal ideal domain)? Prove your answer.
\bigskip




\newpage


\noindent
{\bf Problem 3. (30 points)} Let $\alpha,\beta \in \cee$, both nonzero, let $\gamma = \alpha/\beta$, and suppose $\que(\alpha) \cong \que(\beta)$. 
\medskip

\noindent
{\bf Part a (10 points)} Give necessary and sufficient conditions on $\alpha$ and $\beta$ so that $[\que(\gamma):\que] < \infty$. Prove your answer.
\medskip

\noindent
{\bf Part b (10 points)} Suppose $\alpha$ and $\beta$ are algebraic and $\gamma \in \que$. Let
$$f(x) = \sum_{k=0} a_k x^k$$
be the minimal polynomial of $\alpha$. Determine the minimal polynomial $g(x)$ of $\beta$.
\medskip

\noindent
{\bf Part c (10 points)} Suppose $\alpha$ and $\beta$ are algebraic and $f(x) = g(x)$, i.e. they have the same minimal polynomial. Is it true that~$\gamma \in \que$? Prove or give a counterexample.




\newpage



\noindent
{\bf Problem 4. (20 points)} Is the sequence $\{ \cos n \}_{n = 0}^{\infty}$ dense in the interval $[-1,1]$? Prove your answer.


\newpage



\end{document}