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## 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.

$\newcommand{\aut}{\textrm{Aut}} \newcommand{\sub}{\textrm{Sub}} \newcommand{\join}{\vee} \newcommand{\bigjoin}{\bigvee} \newcommand{\meet}{\wedge} \newcommand{\bigmeet}{\bigwedge} \newcommand{\normaleq}{\unlhd} \newcommand{\normal}{\lhd} \newcommand{\union}{\cup} \newcommand{\intersection}{\cap} \newcommand{\bigunion}{\bigcup} \newcommand{\bigintersection}{\bigcap} \newcommand{\sq}[2][\ ]{\sqrt[#1]{#2\,}} \newcommand{\pbr}[1]{\langle #1\rangle} \newcommand{\ds}{\displaystyle} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\A}{\mathbb{A}} \newcommand{\F}{\mathbb{F}} \newcommand{\T}{\mathbb{T}} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\imp}{\Rightarrow} \newcommand{\rimp}{\Leftarrow} \newcommand{\pinfty}{1/p^\infty} \newcommand{\power}{\mathcal{P}} \newcommand{\calL}{\mathcal{L}} \newcommand{\calC}{\mathcal{C}} \newcommand{\calN}{\mathcal{N}} \newcommand{\calB}{\mathcal{B}} \newcommand{\calF}{\mathcal{F}} \newcommand{\calR}{\mathcal{R}} \newcommand{\calS}{\mathcal{S}} \newcommand{\calU}{\mathcal{U}} \newcommand{\calT}{\mathcal{T}} \newcommand{\gal}{\textrm{Gal}} \newcommand{\isom}{\approx}$

Problem Set 8 Due 03/23/2018 (partial)

1. 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.
2. Prove or disprove: all cyclotomic polynomials have all their coefficients in $\{0,\pm 1\}$.

Problem Set 7 Due 03/16/2018 (complete)

1. Let $E/K$ be an algebraic extension, and let $E_i=E\intersection K^{\pinfty}$. Prove or disprove that $E/E_i$ is separable. (Hint: try first the case $E=\ol{K}$)
2. Each $\varphi\in\aut_K(\ol{K})$ induces a complete lattice automorphism on $\sub_K(\ol{K})$.
3. 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:
1. All elements of $B$ satisfy property $P$ w.r.t. $B$, i.e. $x ∈ B ⇒ P(x,B)$,
2. if $x$ has property $P$ w.r.t. $B$, and $B ⊆ A$, then $x$ has property $P$ w.r.t. $A$, i.e. $(B ⊆ A \textrm{ and } P(x,B))⇒P(x,A)$,
3. 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.
4. Let $E/K$ be an algebraic extension. The normal closure of $E/K$ is the spliting field of the set of polynomials $\displaystyle A = \{{\min}_{K}(\alpha)\mid\alpha\in E^\times\}.$

Problem Set 6 Due 03/09/2018 (complete)

1. Let $K$ be a field of prime characteristic $p$. The field $K^{1/p^\infty}$ is the smallest perfect field that contains $K$.
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 $\beta$. Let $F=K(\alpha,\beta)$. Find $[F:K]$. Show that any $\gamma\in F$ has degree 1 or 2 over $K$.

Problem Set 5 Due 02/23/2018 (complete)

1. Show that a finite subgroup of the multiplicative group $K^\times$ of any field $K$ is cyclic.
2. If $K$ is a perfect field and $F/K$ is an algebraic extension then $F$ is perfect.
3. 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 $K$.
4. 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 $K$.

Problem Set 4 Due 02/16/2018 (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 $F/K$ 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 $\widehat{K}$.
3. Let $\ol{K}$ be an algebraic closure of $K$. Show that $\ol{K}$ is minimal with the property of being an extension of $K$ which is algebraically closed.
4. Let $F/K$ be a field extension and $E,L\in\sub_K(F)$. Show that if $E/K$ is algebraic then $EL$ is algebraic over $L$. If $EL$ is algebraic over $L$, does it follow that $E$ is algebraic over $K$? How about $E/(E\intersection L)$?