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**Hilton Memorial Lecture**

people:fer:504ws:spring2020: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 6** Due 03/31/2020 (complete)

- Show that every $\varphi\in\aut_K\left(\ol{K}\right)$ induces a complete lattice automorphism on $\sub_K\left(\ol{K}\right)$. (This is part of Prop. 4.6.3 in the posted class notes)
- Let $E/K$ be an algebraic extension. Prove that the normal closure of $E/K$ is the spliting field of the set of polynomials \[ A = \left\{{\ds\min_K(\alpha)}\mid\alpha\in E^\times\right\}. \] (This is Prop. 4.7.3 in the posted class notes)
- Let $E/K$ be an infinite separable extension. Prove that $[E:K]_s =_f [E:K]$, meaning both are finite and equal, or both are infinite. (Note that this and its converse were already proved for finite extensions as Prop. 3.71.3 in the posted class notes)
- Find an example of an algebraic extension for which $[E:K]_s =_f [E:K]$, but $E/K$ is NOT separable.

**Problem Set 5** Due 03/24/2020 (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 $K=\F_2(s,t)$ be the field of rational functions in two variables $s$ and $t$, over the two element field, $\F_2$. Let $\alpha=\sqrt{s}$ and $\beta=\sqrt{t}$, i.e. $\alpha$ is a root of $x^2-s\in K[x]$, and similarly for $\beta$. Prove or disprove that $K(\alpha,\beta)$ is a simple extension of $K$.
- Let $K$ be a field of characteristic $p$.
- Show that $K=K^{1/p}$ iff $K$ is perfect.
- Show that the field $K^{1/p^\infty}$ is a perfect field, and the smallest perfect field that contains $K$.

- Let $K$ be a field of characteristic $p$. Is $\ol{K}$ separable over $K^{1/p^\infty}$? Prove or disprove.

**Problem Set 4** Due 03/10/2020(complete)

- Show that the algebraic closure is a
*closure operator*, i.e.- $K\leq\ol{K}$,

- $\ol{\ol{K}} =\ol{K}$,

- $K\leq E \imp \ol{K}\leq\ol{E}$.

- Let $\ol{K}$ be an algebraic closure of $K$. Show:
- $\ol{K}$ is minimal with the property of being an extension of $K$ which is algebraically closed.
- $\ol{K}$ is maximal with the property of being an algebraic extension of $K$.

- Let $\ f(x)\in K[x]$. Prove that if $\alpha$ is a root of $\ f(x)\ $ with multiplicity $m$, then $\alpha$ is a root of $\ f^{(i)}(x)\ $ for all $\ 0\leq i < m$.
- Prove that if $K$ is a perfect field, and $F/K$ is an algebraic extension, then $F$ is a perfect field.

**Problem Set 3** Due 02/25/2020 (complete)

- Prove the corollary stated in class: If $K\leq E_i\leq F$ and each $E_i/K$ is algebraic, then the join $\displaystyle\bigjoin_{i\in I}E_i$ is algebraic over $K$.
- Let $F/K$ be a finite extension. Prove that $\end_K(F)=\aut_K(F)$, i.e. every endomorphism of $F$ that fixes $K$ is an automorphism of $F$.
- Consider the extension $F=\Q(\alpha,\omega)$ of $\Q$ discussed in class, where $\alpha$ is a root of $x^3-2$ and $\omega$ is a root of $x^2+x+1$. Construct several automorphisms of $F$. Is there a bound for the number of automorphisms of $F$?
- Let $F/K$ be a field extension, and $\varphi:F\to L$ a field homomorphism. Let $\widehat{F}=\varphi(F)$ and $\widehat{K}=\varphi(K)$. Prove:
- $[\widehat{F}:\widehat{K}]=[F:K]$.
- If $F/K$ is algebraic, then so is $\widehat{F}/\widehat{K}$.
- If $F/K$ is transcendental, then so is $\widehat{F}/\widehat{K}$.
- If $F$ is an algebraic closure of $K$, then $\widehat{F}$ is an algebraic closure of $\widehat{K}$.

people/fer/504ws/spring2020/homework.txt · Last modified: 2020/03/29 23:51 by fer

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