Department of Mathematics and Statistics, Binghamton University people:fer:504ws:spring2020

Homework

2020-04-30T16:09:26-04:00

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 9 Due 05/05/2020 (complete)

Prove that a finite group $G$ is solvable iff there is a finite sequence of subgroups \[ 1=H_0\leq H_1 \leq \cdots \leq H_{n-1} \leq H_n=G \] such that each $H_i\normaleq H_{i+1}$ and $H_{i+1}/H_i$ is cyclic. Show, with a counterexample, that this equivalence does not hold in general for arbitrary groups.
Show that the class of solvable groups is not closed under arbitrary products.
(Optional) Redo Exercise 4.6.1 in the class notes (page 102)
Let $p$ be prime, and $G\leq S_p$. Show that if $G$ contains a $p$-cycle and a transposition, the $G=S_p$.

Problem Set 8 Due 04/28/2020 (complete)

Prove Theorem 4.24.1,2 in the class notes (page 90).
Exercise 4.6.1 in the class notes (page 101)
Let $K$ and $L$ be fields. Show that the set $\hom(K,L)$ of all homomorphisms from $K$ to $L$, is linearly independent over $L$ as a subset of the vector space $L^K$ of all functions from $K$ to $L$. In particular $\aut(K)$ is linearly independent over $K$.
Let $F/K$ be a finite extension, and $L/K$ its normal closure. Show that $L/K$ is also a finite extension. Hint: if you write $E=K(\alpha_1,\dots,\alpha_n)$, and let $f_i(x)=\min_K(\alpha_i)$, show that $L$ is the splitting field of the set $A=\{f_1(x),\dots,f_n(x)\}$.

Problem Set 7 Due 04/16/2020 (complete)

Prove or disprove: all cyclotomic polynomials have all their coefficients in $\{0,\pm 1\}$.
Show that if $n$ is even then \[ \phi_{2n}(x) = \phi_n(x^2), \] and if $n\geq 3$ is odd then \[ \phi_{2n}(x) = \phi_n(-x). \]
Let $P$ be a locally finite poset. For $y\neq x\in P$, show that \[ \sum_{y\leq \ul{z}\leq x}\mu(z,x)=0 \] Hint: Fix $y\in P$, and then use induction on the Artinian poset \[\{u\in P\mid u > y\}. \]
Show that the sequence of coefficients of the cyclotomic polynomial $\phi_n(x)$, for $n\geq 2$, is palindrome, i.e. if \[ \Phi_n(x)=\sum_{i=0}^{\varphi(n)}a_ix^i, \] then $a_{\varphi(n)-i}=a_i$.

Earlier Homework

2020-04-23T23:12:13-04:00

Math 504 - Earlier Homework

Problem Set 6 Due 04/14/2020 (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:
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 $K$.
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 $p$.
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 $K$.
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.
1. $K\leq\ol{K}$,
2. $\ol{\ol{K}} =\ol{K}$,
3. $K\leq E \imp \ol{K}\leq\ol{E}$.
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 $K$.
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:
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 $\widehat{K}$.

Problem Set 02 Due 02/13/2020 (complete)

Let $F$ be a field extension of $K$, and $a_1,a_2,\dots,a_n\in F$. Prove:
1. $ K[a_1,a_2,\dots,a_n] = K[a_1][a_2]\cdots[a_n]$, and
2. $ K(a_1,a_2,\dots,a_n) = K(a_1)(a_2)\cdots(a_n)$.
Show that $\Q(\sq{2})\neq\Q(\sq{3})$. Generalize.
Grillet, Page 163, IV.2.1
Grillet, Page 163, IV.2.2, IV.2.4

Problem Set 01 Due 02/04/2020 (complete)

Let $G$ be a group and $N\normaleq G$. $G$ is solvable iff $N$ and $G/N$ are solvable. In this case, $$l(G) \leq l(N) + l(G/N).$$
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.
Given a lattice $(L,\join,\meet)$ in the algebraic sense, show that the binary relation $\leq$, defined by $$ x \leq y \quad iff \quad x \meet y = x, $$ is a partial order on $L$. Moreover, for any $x,y \in L$, $x \meet y$ is the $\glb\{x,y\}$, and $x \join y$ is the $\lub\{x,y\}$.
Let $A$ be a universal algebra, and $\sub(A)$ the complete lattice of subuniverses of $A$. If $D\subseteq\sub(A)$ is directed, then $\ds\left(\bigunion_{X\in D}X\right)\in\sub(A)$.

Homework

Spring 2020

2026-04-14T05:10:36-04:00

Algebra II - Math 504 (Spring 2020)

Spring 2020

Syllabus

Instructor:	Fernando Guzmán	WH-116	x-72876	fer@math.binghamton.edu

Office Hours:	Tuesday	4:00 - 5:30
(subject to change)	Thursday	1:00 - 2:30

Announcements:

– Starting on 03/12/2020, this course goes to online mode. Please check the alternative teaching plan for details.

Homework

Posted class notes

Beware that notes with a later date may include edits to earlier notes.

Notes until 04/21/2020
Notes for 04/23/2020 - 04/30/2020
Notes for 05/05/2020