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} $

**Problem Set 6** Due 03/24/2017 (complete)

- Show that if $E/K$ is separable then \mbox{$[E:K]_s =_f [E:K]$}, where $=_f$ means both sides are finite and equal, or both are infinite. Note that this and its converse were proved in class for finite extensions. Show that the converse is not true in general.
- Prove or disprove: all cyclotomic polynomials have all their coefficients in $\{0,\pm 1\}$.
- Let $P$ be a locally finite poset, and $x\neq y\in P$. Show that \[ \sum_{y\leq z\leq x}\mu(y,z)=0 \]

**Problem Set 5** Due 03/10/2017 (complete)

- Let $K\leq E\leq F$, and $\alpha\in F$, algebraic over $K$. Prove:
- If $\alpha$ is separable over $K$, then it is separable over $E$.
- If $\alpha$ is separable over $E$, and $E/K$ is separable, then $\alpha$ is separable over $K$.

- Let $S$ be a set, and $P(x,B)$ denote a property, where $x \in S$ and $B ⊆ S$. When $P(x,B)$ is true, we will say that $x$ has the property $P$, with respect to $B$. For $A,B ⊆ S$, write $P(A,B)$ provided all elements of $A$ have property $P$ w.r.t. $B$, i.e. for all $x∈A$, we have $P(x,B)$. Let $$B^P := \{x\in S\ |\ P(x,B)\}$$ be the set of elements of $S$ related to $B$ via the property $P$. Assume the property $P$ satisfies:
- All elements of $B$ satisfy property $P$ w.r.t. $B$, i.e. $x ∈ B ⇒ P(x,B)$,
- if $x$ has property $P$ w.r.t. $B$, and $B ⊆ A$, then $x$ has property $P$ w.r.t. $A$, i.e. $(B ⊆ A \textrm{ and } P(x,B))⇒P(x,A)$,
- if $x$ has property $P$ w.r.t. $A$, and $P(A,B)$, then $x$ has property $P$ w.r.t. $B$, i.e. $P(x,A) \textrm{ and } P(A,B) ⇒ P(x,B)$.

Show that the map $B \mapsto B^P$ is a closure operator.

- Let $E/K$ be an algebraic extension, and let $E_i=E\intersection K^{\pinfty}$. Prove or disprove that $E/E_i$ is separable.
- Each $\varphi\in\aut_K(\ol{K})$ induces a complete lattice automorphism of $\sub_K(\ol{K})$. All normal extensions of $K$ are fixed points of this automorphism.

people/fer/504ws/504ws_homework.txt · Last modified: 2017/03/21 08:49 by fer

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Noncommercial-Share Alike 3.0 Unported