Homework [Binghamton University Department of Mathematical Sciences]

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{\end}{\textrm{End}} \newcommand{\sub}{\textrm{Sub}} \newcommand{\min}{\textrm{min}} \newcommand{\lub}{\textrm{l.u.b.}} \newcommand{\glb}{\textrm{g.l.b.}} \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)

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 $K$.
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.
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)

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)
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 $K(S)$.
Let $F/E/K$ be a field tower. Prove that \[tr.d._K(F)=tr.d._E(F)+tr.d._K(E)\]
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 $f$)

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Math 504 - Homework

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