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===== Math 504 - Homework =====
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**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 $a_{\varphi(n)-i}=a_i$.
[[people:fer:504ws:Spring2020:old_homework]]
~~META:title=Homework~~
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===== Math 504 - Homework =====
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**Problem Set 13** Due 05/07/2018 (complete)
- Let $K$ and $L$ be fields. Show that the set $\hom(K,L)$ of all homomorphisms from $K$ to $L$, is linearly independent over $L$. In particular $\aut(K)$ is linearly independent over $K$.
- 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.
- Define: an angle $\theta$ is constructible if there are two constructible straight lines forming an angle $\theta$. \\ Prove: let $l$ be a constructible straight line, $A$ a constructible point on $l$, and $\theta$ a constructible angle. The straight line(s) that go through $A$ and form an angle $\theta$ with $l$ is(are) constructible.
**Problem Set 12** Due 04/27/2018 (complete)
- Let $F/K$ be a field extension, $S\subseteq T\subseteq F$ with $S$ algebraically independent over $K$, and $F$ algebraic over $K(T)$. Prove that there is a transcendence basis $B$, for $F$ over $K$, such that $S\subseteq B\subseteq T$. (Hint: prove that a directed union of algebraically independent sets over $K$ is algebraically independent over $K$, and use Zorn's lemma)
- Let $F/K$ be a field extension and $S\subseteq F$. Prove that TFAE:
- $S$ is maximal algebraically independent over $K$,
- $S$ is algebraically independent over $K$ and $F$ is algebraic over $K(S)$,
- $S$ is minimal such that $F$ is algebraic over $K(S)$.
- Let $F/E/K$ be a field tower. Prove that \[tr.d._K(F)=tr.d._E(F)+tr.d._K(E)\]
- Let $K$ be a field, and $t_1,\dots,t_n$ independent variables. If $f(t_1,\dots,t_n)\in K[t_1,\dots,t_n]$ is a symmetric polynomial in variables $t_1,\dots,t_n$, there is a **polynomial** $g$, such that \[ f(t_1,\dots,t_n) = g(s_1,\dots,s_n). \] (Hint: Use double induction on $n$ and $d$, the total degree of $f$)
[[people:fer:504ws:Spring2018:old_homework]]
=== Math 504 - OldHomework =====
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**Problem Set 11** Due 04/20/2018 (complete)
- Let $G$