**Problem of the Week**

**Number Theory Conf.**

**Zassenhaus Conference**

**Hilton Memorial Lecture**

people:fer:504ws:504ws_old_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 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 $G$.

**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.

**Problem Set 4** Due 02/24/2017 (complete)

- Find a field $K$ of characteristic 3, and an irreducible polynomial $p(x)\in K[x]$, such that $p(x)$ is inseparable. What are the multiplicities of each of the roots of $p(x)$?
- Let $K$ be a field of characteristic 0, $f(x)\in K[x]$, $\alpha$ an element of some extension of $K$, and $m\in\N$. Show that the multiplicity of $\alpha$ as a root of $f(x)$ is $\geq m$ iff $\alpha$ is a root of $f^{(i)}(x)$ for all $0\leq i < m$.
- Show that a finite subgroup of the multiplicative group $K^{\times}$ of any field $K$ is cyclic.
- If $K$ is a perfect field, and $F/K$ is an algebraic extension, then $F$ is a perfect field.

**Problem Set 3** Due 02/17/2017 (complete)

- 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 $E/(E\intersection L)$?
- 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 $\widehat{K}$.

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

**Problem Set 2** Due 02/03/2017 (complete)

- Let $A$ be a universal algebra, and $\sub(A)$ the complete lattice of subalgebras of $A$. If $D\subseteq\sub(A)$ is directed, then $\ds\left(\bigunion_{X\in D}X\right)\in\sub(A)$.
- Show that the direct (cartesian) product of two fields is never a field.
- Show that $\Q(\sq{2})\neq\Q(\sq{3})$. Generalize.
- Page 163, IV.2.1
- Page 163, IV.2.2,4

**Problem Set 1** Due 01/27/2017 (complete)

- Let $G$ be a group and $N\normaleq G$. $G$ is solvable iff $N$ and $G/N$ are solvable. In this case, $$l(G) \leq l(N) + l(G/N)$$.
- Given a lattice $(L,\meet,\join)$ in the algebraic sense, show that the binary relation defined by $$ x \leq y \quad iff \quad x \meet y = x $$ is a partial order on $L$, and for $x,y \in L$, $x \meet y$ is the g.l.b.{x,y}, and $x \join y$ is the l.u.b.{x,y}.
- If $L$ is a poset in which every subset has a l.u.b., then every subset of $L$ also has a g.l.b.

people/fer/504ws/504ws_old_homework.txt · Last modified: 2017/04/26 16:52 by fer

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