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people:fer:504ws:spring2020:homework

## 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 6 Due 03/31/2020 (complete)

1. 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)
2. 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)
3. 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)
4. 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)

1. Let $K\leq E\leq F$, and $\alpha\in F$, algebraic over $K$. Prove:
1. If $\alpha$ is separable over $K$, then it is separable over $E$.
2. If $\alpha$ is separable over $E$, and $E/K$ is separable, then $\alpha$ is separable over $K$.
2. 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 $K$.
3. Let $K$ be a field of characteristic $p$.
1. Show that $K=K^{1/p}$ iff $K$ is perfect.
2. Show that the field $K^{1/p^\infty}$ is a perfect field, and the smallest perfect field that contains $K$.
4. 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)

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

2. $\ol{\ol{K}} =\ol{K}$,
3. $K\leq E \imp \ol{K}\leq\ol{E}$.
2. Let $\ol{K}$ be an algebraic closure of $K$. Show:
1. $\ol{K}$ is minimal with the property of being an extension of $K$ which is algebraically closed.
2. $\ol{K}$ is maximal with the property of being an algebraic extension of $K$.
3. 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 $\ 0\leq i < m$.
4. 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)

1. 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 $K$.
2. 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 $F$.
3. 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 $F$?
4. 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:
1. $[\widehat{F}:\widehat{K}]=[F:K]$.
2. If $F/K$ is algebraic, then so is $\widehat{F}/\widehat{K}$.
3. If $F/K$ is transcendental, then so is $\widehat{F}/\widehat{K}$.
4. If $F$ is an algebraic closure of $K$, then $\widehat{F}$ is an algebraic closure of $\widehat{K}$.