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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]]