**Problem of the Week**

**Number Theory Conf.**

**Zassenhaus Conference**

**Hilton Memorial Lecture**

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}} $

**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.

**Problem Set 7** Due 04/07/2017 (complete)

- Prove or disprove: the lattice of centralizers in a group $G$ is a sublattice of $\sub(G)$, the lattice of subgroups of $G$.
- What is the lattice of subgroups of $U_n$? What is the lattice of subfields of the cyclotomic extension $\Q(\xi_n)$? Write down the bijection between these two lattices.
- Show that $\sigma_S$, as defined in class on 03/31/17, is an automorphism of $F/\Q$.
- Show that the Galois group $G$ in McCarthy's Example is isomorphic to $\power(R)$, the power set of $R$, with symmetric difference as the binary operation.
- Let $G$ be a group with identity element $e$. Let $\calB_e$ be a collection of subgroups of $G$ which form a basis for the neighborhoods of $e$. Show that the collection \[ \{gH|\ g\in G, H\in\calB_e\}, \] of all left cosets of the subgroups in $\calB_e$ is a basis for a topology on $G$.

people/fer/504ws/504ws_homework.txt · Last modified: 2017/04/17 12:33 by fer

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