\documentclass[12pt]{amsart}
\title{Algebra 2. Homework 9}
\author{}

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\begin{document}
\maketitle

\problem{1} Let $K$ be a field and let $p(x)\in K[x]$ be a monic polynomial of
degree $n$ polynomial.
Let $L/K$ be the splitting extension of $p(x)$. Prove that $[L:K]$
divides $n!$.\\

\problem{2} Prove that $f(x)=x^6+x^3+1\in\Q[x]$ is irreducible.
Determine its $6$ roots in $\C$.\\

\problem{3} Prove that the splitting extension of $x^3-2$ over $\Q$
has degree $6$.\\

\problem{4} Let $L/K$ be an algebraic extension. Let $\sigma: L\to L$ be a 
ring homomorphism
such that $\left. \sigma \right|_{K} = \Id_K$. Prove that $\sigma$
is then an isomorphism.\\

\problem{5} Give an example to prove that the assertion of Problem 1 is false, if
$E/F$ is not assumed to be algebraic.\\

\problem{6} Let $\zeta = \exp\lp \frac{2\pi\iota}{7}\rp \in \C$. Prove that
$\Q(\zeta)$ is a Galois extension of $\Q$.\\

\problem{7} Let $k\subset K$ be two fields, and assume that $K$ is algebraically closed.
Let $K'$ consist of all elements of $K$ which are algebraic over $k$. Prove
that $K'$ is isomorphic to the algebraic closure of $k$.\\

\problem{8} Let $a,b\in\Q$ and consider $f(x) = x^3-ax+b\in\Q[x]$.
Let $L$ be the splitting extension of $f(x)$ over $\Q$.
If $\alpha,\beta,\gamma\in L$ are the roots of $f$, then set
$\Delta=(\alpha-\beta)^2(\alpha-\gamma)^2(\beta-\gamma)^2\in\Q$ 
($(-1)$ times the discriminant of $f$, as defined in Lecture 27).
Assuming $f(x)$ is irreducible, prove that
$[L:\Q]=6$ if and only if $\sqrt{\Delta}\not\in\Q$.\\

\problem{9} Let $K$ be a field
and let $f(x),g(x)\in K[x]$ be two monic polynomials of degrees
$m,n$ respectively. Let $L/K$
be the splitting extension of $\{f(x),g(x)\}$. Let $r_1,\ldots,r_m;
s_1,\ldots,s_n\in L$ be the roots of $f(x)$ and $g(x)$ respectively.

\begin{enumerate}
\item Prove that the following is an element of $K$:
\[
\mathrm{Res}(f,g) = \prod_{\begin{subarray}{c} 1\leq i\leq m \\ 1\leq j\leq n
\end{subarray}} (r_i-s_j),
\]
{\em called the resultant of $f$ and $g$}.\\

\item Prove that there is a unique polynomial $P(A_1,\ldots,A_m; B_1,\ldots,B_n)$
in $m+n$ variables
such that, for every $f(x)=x^m+\sum_{j=1}^m a_j x^{m-j}$ and $g(x)
=x^n+\sum_{k=1}^n b_k x^{n-k}$, $\mathrm{Res}(f,g)=P(a_1,\ldots,a_m;b_1,\ldots,b_n)$.\\

\item Prove that $\mathrm{Res}(f,g)=0$ if and only if $f$ and $g$
have a common root in $\ol{K}$ (algebraic closure of $K$).\\

\item For $f=x^2+b_1x+c_1$ and $g=x^2+b_2x+c_2$, compute $\mathrm{Res}(f,g)$.\\
\end{enumerate}

\problem{10} Let $K$ be a field and let $L=K(T)$ be the field of
rational functions in one variable $T$. 
\begin{enumerate}
\item Prove that for every
$\ds g=\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)
\in \mathrm{GL}_2(K)$, we have an element $\sigma_g \in
\Gal{L}{K}$ given by:
\[
\sigma_g(T) = \frac{aT+b}{cT+d}.
\]

\item Let $\psi : \mathrm{GL}_2(K) \to \Gal{L}{K}$ be given by $\psi(g)=\sigma_g$.
Verify that $\psi$ is a group homomorphism and $\Ker(\psi) = K^{\times}\Id_2$,
where $K^\times=K\setminus\{0\}$ and $\Id_2$ is the $2\times 2$ identity matrix.\\

\item Let $\mathrm{PGL}_2(K) = \mathrm{GL}_2(K)/\Ker(\psi)$. Prove
that $\mathrm{PGL}_2(K)\cong \Gal{L}{K}$.\\

\item Assume that $K$ is an infinite field. Prove that $L/K$ is a Galois extension.\\
\end{enumerate}

(Hint for (3). Let $\ds y=\frac{g(T)}{h(T)} \in K(T)$, where $g$ and $h$
have no common factor. Define $\mathrm{ht}(y) := \max(\deg(g),\deg(h))$, height of $y$.
Prove that, if $y\not\in K$, then $g(X)-h(X)y \in (K(y))[X]$ is irreducible.
Use this to show that, if $\mathrm{ht}(y)=n>0$, then $L$ is an algebraic
extension of $K(y)$ of degree $n$.)

\end{document}
