\documentclass{amsart}
\title{Algebra 2. SAMPLE Final}
\author{}

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\begin{document}
\maketitle
\thispagestyle{empty}

\problem{1} Let $R$ be a commutative ring (with $1\neq 0$). Prove that $M\otimes - :
\Rmod\to \Rmod$ is right exact. Give an example when it is not left exact.\\

\problem{2} State and prove Baer's criterion for injectivity.\\

\problem{3} Let $K$ be a field and $R = K[x,y]$. Construct a free resolution
of $K$ viewed as an $R$--module, where $x$ and $y$ act as zero. That is,
$K\cong R/(x,y)$ as $R$--module.\\

\problem{4} Let $L/K$ be a finite Galois extension. Let $G = \gal{E}{F}$.
Prove that for every group homomorphism $\chi : G \to F^{\times}$, there exists
$\alpha\in E^{\times}$, such that 
$\displaystyle\chi(\sigma) = \frac{\alpha}{\sigma(\alpha)}$ 
for every $\sigma\in G$.\\
({\em You may use Dedekind's theorem about independence of characters, without
proving it.})\\

\problem{5} Let $K$ be a field containing a primitive $n^{\sst{\text{th}}}$
root of unity. Let $L = K(\alpha)$, with $\alpha^n = a\in F$, be a cyclic
extension of degree $n$. Let $\beta\in L$ be any element such that $\beta^n
\in F$. Prove that, there exist some $c\in F$ and $k\in \{1,\ldots,n\}$,
such that $\beta = c\alpha^k$.\\

\problem{6} State (without proof) the necessary and sufficient conditions
for a field $K$ to be imperfect. Prove that every finite field is perfect.\\

\problem{7} Let $\displaystyle \alpha = \left( 3+\sqrt{7}\right)^{\frac{1}{3}}$. Prove
that $\mathbb{Q}(\alpha)$ is not a Galois extension over $\mathbb{Q}$.
\end{document}

