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\begin{document}
\title{MATH 195, Spring 2015, Take-home midterm}

\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, 3/31/15, 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. Good luck!}
\bigskip

\noindent
{\bf Problem 1. (10 points)} Let $n$ be a positive integer. A complex number
$$\zeta_n = e^{\frac{2\pi i}{n}}$$
is called {\it $n$-th primitive root of unity}. Notice that $\zeta_n^n = e^{2\pi i} = 1$ by Euler's formula, and hence $\zeta_n$ is algebraic: it is a root of the polynomial $x^n-1$.
\bigskip

Let $n \geq 2$, and consider the following argument. First notice that
$$\ln \zeta_n = \frac{2\pi i}{n}.$$
On the other hand,
$$0 = \ln 1 = \ln \zeta_n^n = n \ln \zeta_n,$$
and hence $\ln \zeta_n = 0$. But obviously $\frac{2\pi i}{n} \neq 0$. What went wrong? Explain your answer.


\newpage

\noindent
{\bf Problem 2. (10 points)} Let $K=\que(\sqrt{2},\sqrt{3})$. Find an integral primitive element for $K$, i.e. an element $\alpha \in \OO_K$ such that $K=\que(\alpha)$. Prove your answer.

\newpage



\noindent
{\bf Problem 3. (20 points)} A subset $S$ of $\real$ is called {\it discrete} if there exists real $\eps > 0$ such that for every two distinct elements $\alpha,\beta \in S$, 
$$|\alpha - \beta| \geq \eps.$$
On the other hand, let us say that $S$ is {\it near-discrete} if for every $\alpha \in S$ there exist a real $\eps = \eps(\alpha) > 0$ such that
$$|\alpha - \beta| \geq \eps$$
for every $\beta \in S$ distinct from $\alpha$.
\bigskip


{\bf Part 1. (10 points)} Prove that every discrete subset of $\real$ is countable.
\smallskip

{\bf Part 1. (10 points)} Prove that every near-discrete subset of $\real$ is countable.


\newpage



\noindent
{\bf Problem 4. (20 points)} Let $\mu > 0$. We say that $p/q \in \que$ is a $\mu$-approximation to the real number $\alpha$ if
$$\left| \alpha - \frac{p}{q} \right| \leq \frac{1}{q^{\mu}}.$$
Let $S \subseteq \real$ be a set with the following properties:
\begin{enumerate}
\item Every element of $S$ has infinitely many rational $3$-approximations.
\item If a rational number $p/q$ is a $3$-approximation for some $\alpha \in S$, then it is not a $3$-approximation for any other element of $S$.
\end{enumerate}
\bigskip

{\bf Part 1. (10 points)} Prove that $S$ is countable.
 \smallskip

{\bf Part 2. (10 points)} Prove that every $\alpha \in S$ is transcendental.
 
\newpage


\noindent
{\bf Problem 5. (20 points)} Let $K$ be a subfield of $\cee$. An embedding of $K$ into $\cee$ is an injective field homomorphism $\tau : K \to \cee$.
\bigskip

{\bf Part 1. (10 points)} Suppose that $\tau : K \to \cee$ is a field homomorphism such that for some $a \in K$, $\tau(a) \neq 0$. Prove that $\tau$ is an embedding.
\smallskip

{\bf Part 2. (10 points)} Let $K=\que(\alpha)$ for some $\alpha \in \cee$. Prove that $\alpha$ is transcendental if and only if there exist infinitely many distinct embeddings of $K$ into $\cee$.


\newpage


\noindent
{\bf Problem 6. (30 points)} Let $R$ be a subring of the ring $S$, both commutative rings with identity. Let $\alpha \in S$, and define
$$R[\alpha] = \left\{ f(\alpha) : f(x) \in R[x] \right\}.$$
More generally, for $\alpha_1,\dots,\alpha_n \in S$ define recursively
$$R[\alpha_1,\dots,\alpha_n] = R_{n-1}[\alpha_n],$$
where $R_{n-1}=R[\alpha_1,\dots,\alpha_{n-1}]$. A subring $T$ of $S$ is called a finitely generated ring extension of $R$ if $T=R[\alpha_1,\dots,\alpha_n]$ for some $\alpha_1,\dots,\alpha_n \in S$.
\bigskip

{\bf Part 1. (5 points)} Prove that $R[\alpha]$ is a subring of $S$.
\smallskip

{\bf Part 2. (5 points)} Prove that $R[\alpha]$ is the smallest ring containing $R$ and $\alpha$, with respect to inclusion.
\smallskip

{\bf Part 3. (10 points)} Let $T=\zed[1/n]$ for some integer $n > 1$. Compute the group of units $T^{\times}$ of $T$.
\smallskip

{\bf Part 4. (10 points)} Prove that $\que$ is not a finitely generated ring extension of $\zed$.

\newpage


\noindent
{\bf Problem 7. (20 points)} Let $\{n_j\}_{j=1}^{\infty}$ be a sequence of natural numbers such that
$$\lim_{j \to \infty} \frac{n_{j+1}}{n_j} = \infty.$$
Define
$$f(x) = \sum_{j=1}^{\infty} x^{n_j},$$
and let $\alpha$ be a real algebraic number such that $0 < \alpha < 1$. Prove that $f(\alpha)$ is transcendental.


\end{document}