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## Math 504 - Homework

• LaTeX-ed solutions are encouraged and appreciated.
• If you use LaTeX, hand-in a printed version of your homework.
• You are encouraged to discuss homework problems with classmates, but such discussions should NOT include the exchange or written material.
• Writing of homework problems should be done on an individual basis.
• Outside references for material used in the solution of homework problems should be fully disclosed.
• References to results from the textbook and/or class notes should also be included.
• The following lists should be considered partial and tentative lists until the word complete appears next to it.
• Use 8.5in x 11in paper with smooth borders. Write your name on top of each page. Staple all pages.

$\newcommand{\aut}{\textrm{Aut}} \newcommand{\sub}{\textrm{Sub}} \newcommand{\join}{\vee} \newcommand{\bigjoin}{\bigvee} \newcommand{\meet}{\wedge} \newcommand{\bigmeet}{\bigwedge} \newcommand{\normaleq}{\unlhd} \newcommand{\normal}{\lhd} \newcommand{\union}{\cup} \newcommand{\intersection}{\cap} \newcommand{\bigunion}{\bigcup} \newcommand{\bigintersection}{\bigcap} \newcommand{\sq}[2][\ ]{\sqrt[#1]{#2\,}} \newcommand{\pbr}[1]{\langle #1\rangle} \newcommand{\ds}{\displaystyle} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\A}{\mathbb{A}} \newcommand{\F}{\mathbb{F}} \newcommand{\T}{\mathbb{T}} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\imp}{\Rightarrow} \newcommand{\rimp}{\Leftarrow} \newcommand{\pinfty}{1/p^\infty} \newcommand{\power}{\mathcal{P}} \newcommand{\calL}{\mathcal{L}} \newcommand{\calC}{\mathcal{C}} \newcommand{\calN}{\mathcal{N}} \newcommand{\calB}{\mathcal{B}} \newcommand{\calF}{\mathcal{F}} \newcommand{\calR}{\mathcal{R}} \newcommand{\calS}{\mathcal{S}} \newcommand{\calU}{\mathcal{U}} \newcommand{\calT}{\mathcal{T}} \newcommand{\gal}{\textrm{Gal}} \newcommand{\isom}{\approx} \renewcommand{\hom}{\textrm{Hom}}$

Problem Set 13 Due 05/07/2018 (complete)

1. Let $K$ and $L$ be fields. Show that the set $\hom(K,L)$ of all homomorphisms from $K$ to $L$, is linearly independent over $L$. In particular $\aut(K)$ is linearly independent over $K$.
2. Prove that a finite group $G$ is solvable iff there is a finite sequence of subgroups $1=H_0\leq H_1 \leq \cdots \leq H_{n-1} \leq H_n=G$ such that each $H_i\normaleq H_{i+1}$ and $H_{i+1}/H_i$ is cyclic. Show, with a counterexample, that this equivalence does not hold in general for arbitrary groups.
3. Define: an angle $\theta$ is constructible if there are two constructible straight lines forming an angle $\theta$.
Prove: let $l$ be a constructible straight line, $A$ a constructible point on $l$, and $\theta$ a constructible angle. The straight line(s) that go through $A$ and form an angle $\theta$ with $l$ is(are) constructible.

Problem Set 12 Due 04/27/2018 (complete)

1. Let $F/K$ be a field extension, $S\subseteq T\subseteq F$ with $S$ algebraically independent over $K$, and $F$ algebraic over $K(T)$. Prove that there is a transcendence basis $B$, for $F$ over $K$, such that $S\subseteq B\subseteq T$. (Hint: prove that a directed union of algebraically independent sets over $K$ is algebraically independent over $K$, and use Zorn's lemma)
2. Let $F/K$ be a field extension and $S\subseteq F$. Prove that TFAE:
1. $S$ is maximal algebraically independent over $K$,
2. $S$ is algebraically independent over $K$ and $F$ is algebraic over $K(S)$,
3. $S$ is minimal such that $F$ is algebraic over $K(S)$.
3. Let $F/E/K$ be a field tower. Prove that $tr.d._K(F)=tr.d._E(F)+tr.d._K(E)$
4. Let $K$ be a field, and $t_1,\dots,t_n$ independent variables. If $f(t_1,\dots,t_n)\in K[t_1,\dots,t_n]$ is a symmetric polynomial in variables $t_1,\dots,t_n$, there is a polynomial $g$, such that $f(t_1,\dots,t_n) = g(s_1,\dots,s_n).$ (Hint: Use double induction on $n$ and $d$, the total degree of $f$)