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--- people:fer:504ws:spring2020:homework [2020/02/20 12:07]
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+~~META:title=Homework~~
+<WRAP centeralign>
+===== Math 504 - Homework =====
+</WRAP>
+{{page>people:fer:504ws:504ws_homework_header&nofooter&noeditbtn}}
+----
+{{page>people:fer:504ws:defs&nofooter&noeditbtn}}
+**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]]

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