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people:fer:504ws:504ws_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{\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} $

**Problem Set 10** Due 05/08/2017 (complete)

- Complete the proof of the following proposition. The first part was done in class. If $E/K$ is a Galois extension, and $F/K$ is any field extension, then $EF/F$ is a Galois extension. Moreover, $\gal(EF/F)$ embeds in $\gal(E/K)$, and when $E/E\intersection F$ is a finite extension, \[ \gal(EF/F) \isom \gal(E/E\intersection F).\]
- Show 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, by counterexample, that the
**finite**hypothesis in the previous problem is necessary. - Let $l$ be a constructible straight line, $A$ a constructible point on $l$, and $\theta$ a constructible angle. Show that the straight line that goes through $A$ and forms an angle $\theta$ with $l$ is constructible.

**Problem Set 9** Due 05/01/2017 (complete)

- Prove that a directed union of algebraically independent sets over $K$ is algebraically independent over $K$. In particular, the union of a chain of algebraically independent sets over $K$ is algebraically independent over $K$.
- Given $S\subseteq T$ with $S$ algebraically independent over $K$ and $F$ algebraic over $K(T)$, there is a transcendence basis $B$ with $S\subseteq B\subseteq T$. In particular, any field extension $F/K$ has a transcendence basis.
- Prove the following version of the exchange property: Let $F/K$ be a field extension, $S,T\subseteq F$ be each algebraically independent over $K$, with $|S| < |T|$. There is $\beta\in T-S$ such that $S\union \{\beta\}$ is algebraically independent over $K$.
- Prove that for a tower $L/F/K$, \[ tr.d._K(L) = tr.d._F(L) + tr.d._K(F) \]
- Prove that if $f(t_1,\dots,t_n)$ is a symmetric
**polynomial**in variables $t_1,\dots,t_n$, there exists a**polynomial**$g$ such that $f(t_1,\dots,t_n)=g(s_1,\dots,s_n)$.

**Problem Set 8** Due 04/21/2017 (complete)

- Let $G$ be a group, and $H_1,H_2\leq G$. Show that \[ [G:H_1\intersection H_2] \leq [G:H_1][G:H_2]. \]
- Prove that the normal closure of a finite separable extension over $K$ is a finite Galois extension.
- Given a projective system $(G_i|i\in I)$ of groups, with maps $(\rho_{i,j}|i\leq j)$, show that the following subset of the product \[ \left\{a\in\prod_{i\in I}G_i\middle|\rho_{i,j}(a_j)=a_i \text{ for all } i\leq j\right\}, \] together with the projections on the factors is a projective limit for the system.

people/fer/504ws/504ws_homework.txt · Last modified: 2017/05/04 11:04 by fer

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