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\begin{document}
\title{Math 200b Winter 2022 Homework 7}
\date{Complete before final exam, but no need to hand in}
\maketitle


1.  (a).
Suppose that $K/F$ is a Galois extension with $[K:F] = p^n$ for some prime $p$ and $n \geq 1$.  Show that 
there are intermediate fields $F \subseteq E_1 \subseteq K$, $F \subseteq E_2 \subseteq K$ such that 
$E_i/F$ is Galois for $i = 1,2$ and $[E_1:F] = p$, $[E_2:F] = p^{n-1}$.

(b).  Suppose that $f \in \mb{Q}[x]$ is irreducible of degree $4$ over $\mb{Q}$, and that its splitting field $K$ over 
$\mb{Q}$ satisfies $\operatorname{Gal}(K/\mb{Q}) \cong S_4$.  Show that there is a field $\mb{Q} \subseteq E \subseteq K$ such that $[E:\mb{Q}] = 4$ and there are no proper intermediate fields between $\mb{Q}$ and $E$.

\bigskip


2.  Let $p$ be prime and let $\mb{F}_{p^n}$ be a field with $p^n$ elements.
Let $S$ be the set of generators (as a group) of the multiplicative group $(\mb{F}_{p^n})^*$.


\medskip

(a).  Let $f \in \mb{F}_p[x]$ be an irreducible polynomial of degree $n$.  Show that 
$f$ splits in $\mb{F}_{p^n}$ and that either all of its roots are in $S$ or none of them is.  (hint:  for any two roots of $f$, there is an automorphism sending one to the other).

\medskip

(b).  Show that $n \, | \, \varphi(p^n -1)$ for all primes $p$ and all $n \geq 1$, where $\varphi$ is the Euler phi-function.  (hint: use part (a)).

\medskip

(c).  Consider the explicit case of the field $\mb{F}_{16}$.  Find all irreducible polynomials of degree $4$ over $\mb{F}_2$.  Which ones have roots in $S$?  

\bigskip




3.  Let $\zeta \in \mb{C}$ be a primitive $p$th root of $1$ for some prime $p \geq 3$.   Let $K = \mb{Q}(\zeta)$ be the splitting field of $x^p-1$ inside $\mb{C}$.

\medskip

(a).   Let $\alpha = \sum_{i=0}^{p-1} \zeta^{i^2}$.  
This is called a \emph{Gauss sum}.  Prove that $E = \mb{Q}(\alpha)$ is the unique subfield of $K$ such that $[E:\mb{Q}] = 2$.  

\medskip

(b).   Show that $L = \mb{Q}(\zeta + \zeta^{-1})$ is the unique subfield of $K$ such that $[K:L] = 2$.  
Show that in fact $L = K \cap \mb{R}$.  (Hint:  note that complex conjugation restricts to an automorphism of $K$).

\bigskip

\bigskip


4.  Let $f = x^p -2$ for some prime $p \geq 3$.  Consider the splitting field $K$ of $f$ over $\mb{Q}$.  Show that $K/\mb{Q}$ is Galois with $[K:\mb{Q}] = p(p-1)$.  Prove that 
$G = \Gal(K/\mb{Q})$ is isomorphic to the semidirect product $\mb{Z}_p^* \ltimes_{\psi} \mb{Z}_p$, where $\psi: \mb{Z}_p^* \to \on{Aut}(\mb{Z}_p)$ is the natural isomorphism. 








\end{document}
