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Math 504 - 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} $

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

  1. 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 $p(x)$?
  2. 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 $0\leq i < m$.
  3. Show that a finite subgroup of the multiplicative group $K^{\times}$ of any field $K$ is cyclic.
  4. 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)

  1. 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 $E/(E\intersection L)$?
  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. Show that the algebraic closure is a closure operator, i.e.
    1. $K\leq\ol{K}$,
    2. $\ol{\ol{K}} =\ol{K}$,
    3. $K\leq E \imp \ol{K}\leq\ol{E}$.

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

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

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

  1. Let $G$ be a group and $N\normaleq G$. $G$ is solvable iff $N$ and $G/N$ are solvable. In this case, $$l(G) \leq l(N) + l(G/N)$$.
  2. 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}.
  3. 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: 2017/03/13 09:21 by fer