\documentclass[12pt]{amsart}
\title{Algebra 2. Homework 10}
\author{}

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\begin{document}
\maketitle

\problem{1} Prove that every algebraic extension of a perfect field
is perfect.\\

\problem{2} Let $p\in \Z_{\geq 2}$ be a prime number and $\FF_p$
be the field with $p$ elements. Let $\lambda$ be a variable and let
$K=\FF_p(\lambda)$ be the field of rational functions in $\lambda$
with coefficients from $\FF_p$.

For each $n\in\Z_{\geq 1}$, let $K_n$ be the splitting extension
of $x^{p^n}-\lambda \in K[x]$.

\begin{enumerate}
\item Prove that we have a chain of field extensions:
\[
K\subset K_1\subset K_2 \subset \ldots
\]

\item Prove that $\Gal{K_n}{K} = \{\Id\}$ for every $n$.

\item Let $\ds \wt{K} = \bigcup_{n\geq 1} K_n$. Prove that $\wt{K}$
is a perfect field.\\
\end{enumerate}

\problem{3} Again, let $p\in\Z_{\geq 2}$ be a prime number. Consider
$K=\FF_p(\lambda,\mu)$ and $L$ be the splitting extension of
$\{x^p-\lambda,x^p-\mu\}\subset K[x]$. Prove or disprove: there exists
$\gamma\in L$ such that $L\cong K(\gamma)$.\\

\problem{4} Let $L/K$ be a Galois extension. Let $\alpha\in L$
and $f(x)=\minpoly{\alpha}{x}\in K[x]$. Let $n=\deg(f)\in\Z_{\geq 1}$.
Let $\{r_1,\ldots,r_n\}\subset L$ be the set of $n$ distinct roots
of $f(x)$.

\begin{enumerate}
\item Prove that we have a group homomorphism
\[
\phi_f : \Gal{L}{K} \to \Sym_n,
\]
given by $\phi_f(g) = \sigma$ if $g(r_i) = r_{\sigma(i)}$ for every
$1\leq i\leq n$.

\item Let $E=K(r_1,\ldots,r_n)\subset L$. Prove that the restriction
homomorphism $\rho_{E,L}:\Gal{L}{K} \to \Gal{E}{K}$ commutes with
$\phi_f$. That is, the following diagram commutes:

\[
\xy
(0,0)*{\Gal{L}{K}}="lk";
(30,0)*{\Gal{E}{K}}="ek";
(15,-10)*{\Sym_n}="sg";
{\ar^{\rho_{E,L}} "lk"; "ek"};
{\ar_{\phi_f} "lk"; "sg"};
{\ar^{\phi_f} "ek"; "sg"};
\endxy
\]

\item Prove that the action of $\Gal{E}{K}$ 
on the set $\{1,\ldots,n\}$, via $\phi_f$, is transitive. That is, for any $i,j$
there exits $g\in \Gal{E}{K}$ such that $\phi_f(g):i\mapsto j$.\\
\end{enumerate}

\problem{5} Let $L/K$ be a normal extension. Let $E$ be an intermediate field:
$K\subset E\subset L$ and let $\sigma: E\to L$ be an embedding which is the identity
on $K$. Prove that $\sigma$ extends to an automorphism of $L$.\\

\problem{6} Let $K = \Q$, $E$ be the splitting field of $X^2-2\in \Q[X]$ over $\Q$.
Thus $E = \Q(\sqrt{2})$.
Let $L$ be the splitting field of $X^2-\sqrt{2}\in E[X]$ over $E$. Describe the
group $\Gal{L}{K}$. Prove that $L/K$ is not a Galois extension.
{\em Splitting extension of a splitting extension is not necessarily
splitting extension.}\\

\problem{7} Let $L = \Q(\sqrt{2},\sqrt{3})$ and $K = \Q$. Compute $\Gal{L}{K}$.
Is $L/K$ a Galois extension?\\

\problem{8} Let $\alpha = \sqrt{2+\sqrt{2}} \in \R$. 
Compute $\Gal{\Q(\alpha)}{\Q}$.\\

\problem{9} Let $n\in\Z_{\geq 2}$ and consider $L=\Q(T_1,\ldots,T_n)$.
Let $G$ be a subgroup of $\Sym_n$, and let $K=L^G$.
Prove that $L/K$ is a Galois extension with
$\Gal{L}{K} = G$.\\

\hrule

\vspace*{0.2in}

Assume that $L/K$ is a Galois extension and let $\Gamma=\Gal{L}{K}$. 
Given an $L$--vector space $V$,
a $K$--{\em structure} on $V$ is the data of a vector sub-$K$-subspace
$V_0\subset V$ such that the canonical map $L\otimes_K V_0 \to V$,
given by $\alpha\otimes v \mapsto \alpha\cdot v$,
is an isomorphism.

Let $V_0\subset V$ be a $K$--structure on an $L$--vector space $V$.
We get an action of $\Gamma$ on $V$ as follows. Let $\{e_i\}_{i\in I}$
be a basis of $V_0$. For any $v\in V$, write $v=\sum_{i\in I} \alpha_i e_i$,
where $\alpha_i\in L$. Then, for any $\sigma\in\Gamma$ we define:
\[
u_{\sigma}(v) = \sum_{i\in I} \sigma(\alpha_i) e_i.
\]

\problem{10} Let $L/K$, $V_0\subset V$, $\Gamma$
and $\{u_\sigma : V\to V\}_{\sigma\in\Gamma}$ be as above.

\begin{enumerate}
\item Prove that $\sigma\mapsto u_\sigma$ is a group homomorphism
$\Gamma \to \Aut_{K\!-\!\scriptstyle{vs}}(V)$ such that for every $\alpha\in L$,
$\sigma\in\Gamma$ and $v\in V$, we have:
\[
u_\sigma(\alpha v) = \sigma(\alpha) u_\sigma(v).
\]

\item Let $x\in V$. Prove that $x\in V_0$ if and only if $u_\sigma(x)=x$ for
every $\sigma\in \Gamma$.

\item Let $U\subset V$ be a vector sub-$L$-subspace. Prove that $U_0 = U\cap V_0$
is a $K$--structure on $U$ if and only if $u_\sigma(U)\subset U$ for every
$\sigma\in\Gamma$. (In this case, we say $U$ is rational over $K$).

\item Let $V,W$ be two $L$--vector spaces, together with $K$--structures $V_0\subset V$
and $W_0\subset W$. Let $f:V\to W$ be an $L$--linear map.
Prove that $f(V_0)\subset W_0$ if and only if $f(u_\sigma(x)) = u_{\sigma}(f(x))$
for every $x\in V$. (In this case, we say $f$ is rational over $K$).\\
\end{enumerate}

\problem{11} Let $V$ be an $L$--vector space and assume that we are given
a group homomorphism $u:\Gamma \to \Aut_{K\!-\!\scriptstyle{vs}}(V)$
(denoted as before by $\sigma\mapsto u_\sigma$) such that
\[
u_\sigma(\alpha v) = \sigma(\alpha)u_\sigma(v),\qquad
\forall\ \sigma\in\Gamma,\ \alpha\in L, v\in V.
\]
Let $V_0 = \{v\in V : u_\sigma(v)=v,\ \forall\ \sigma\in\Gamma\}$. Prove
that if $|\Gamma|<\infty$ then $V_0\subset V$ is a $K$--structure on $V$.\\

\hrule

\vspace*{0.2in}

\problem{12} (from Algebra 1). Let $A=\Z$, or $K[x]$. Let $M$ be a finitely
generated, torsion module over $A$. Prove that there exists $x\in M$
such that $\mathrm{Ann}(x) = \mathrm{Ann}(M)$. Here,
\[
\mathrm{Ann}(x)=\{a\in A : ax=0\},\qquad \mathrm{Ann}(M)
=\{a\in A : am=0,\ \forall\ m\in M\}.
\]
\end{document}
