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+~~META:title=Homework~~
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+===== Math 402 - 01 Homework (Spring 2019)=====
+</WRAP>
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+----
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+**Problem Set 10** (complete) Due: 05/10/2019
+  - Let  $F$  be a field of characteristic zero,  $a\in F$ , and  $\xi=\xi_n$  a primitive  $n$ -th root of unity.
+    - Show by example that  $\gal_F(F(\xi))$  need not be all of  $U_n$ .
+    - Show by example that  $\gal_{F(\xi)}(F(\xi,​\sq[n]{a}))$  need not be all of  $C_n$ .
+  - Let  $G$  and  $H$  be solvable groups.  Prove that  $G\times H$  is solvable.
+  -   Show that the change of variable  $y=x+(a/3)$  transforms the general cubic equation  $x^3+ax^2+bx+c = 0$  into a depressed cubic.  Therefore, Cardano's formula is useful to solve any cubic equation.
+**Problem Set 09** (complete) Due: 05/03/2019  Board presentation: 05/10/2019
+  - Prove that the homomorphism    $\begin{array}{rccc} \psi:& U_n &​\to ​    &​\gal(\Q(\xi_n)/​\Q) \\         & ​ k  &​\mapsto &\psi_k \\ \end{array}$  is surjective and injective.
+  - Let  $\xi_{15}=\cis(2\pi/​15)$  be a primitive  $15$ -th root of unity.
+    - Find the group  $\gal(\Q(\xi_{15})/​\Q)$  and draw its lattice of subgroups.
+    - Find and draw the lattice of intermediate fields of the extension  $\Q(\xi_{15})/​\Q$ .
+    - Write down the correspondence between the subgroups in part 1, and the subfields in part 2, using the Fundamental Theorem of Galois Theory.
+  - Show that any non-abelian simple group is non-solvable.
+  - Show that if  $d$  is a divisor of  $n$  then  $\Q(\xi_d)$  is a subfield of  $\Q(\xi_n)$ .  Conclude that  $\varphi(d)$  divides  $\varphi(n)$ , and  $U_d$  is a quotient of  $U_n$ .
+**Problem Set 08** (complete) Due: 04/26/2019  Board presentation: 05/03/2019
+  - Prove the following corollary to the Fundamental Theorem of Galois Theory.  Use only the FTGT statements to prove it.    Let  $E/F$  be a (finite) Galois extension, with Galois group  $G=\gal_F(E)$ .    Let  $L_1,​L_2\in\sub_F(E)$  and  $H_1,​H_2\in\sub(G)$ .
+    -   $(L_1\meet L_2)^* = L_1^* \join L_2^*$
+    -   $(L_1\join L_2)^* = L_1^* \meet L_2^*$
+    -   $(H_1\meet H_2)^* = H_1^* \join H_2^*$
+    -   $(H_1\join H_2)^* = H_1^* \meet H_2^*$
+  - Let  $f(x)\in\Q[x]$  be such that it has a non-real root.  Let  $E$  be the splitting field of  $f(x)$  over  $\Q$ .  Prove that  $\gal_\Q(E)$  has even order.
+  - Consider the polynomial  $f(x)=x^3+2x^2+2x+2\in\Q[x]$ , and  $E$  its splitting field over  $\Q$ .
+    - Show that  $f(x)$  has exactly one real root. (Hint: use calculus)
+    - Show that  $f(x)$  is irreducible over  $\Q$ .
+    - Find  $[E:​\Q]$ .  Fully explain your calculation.
+    - Determine  $\gal_\Q(E)$ .
+  - Consider the group  $S_n$  of all permutations of the set  $\{1,​2,​\dots,​n\}$ .
+    - Show that the transpositions  $(1\ \ 2),(2\ \ 3),​\dots,​(n-1\ \ n)$  generate the whole group  $S_n$ .
+    - Show that  $S_n$  is generated by the following two permutations:  $\rho = (1\ \ 2\ \ \dots\ \ n) \quad \text{and} \quad \sigma=(1\ \ 2)$  (Hint: conjugate  $\sigma$  by  $\rho$ .)
+    - For  $p$  is a prime,  $\rho$  a  $p$ -cycle, and  $\sigma$  a transposition, show that  $\rho$  and  $\sigma$  generate  $S_p$ .  Show, by counterexample, that the hypothesis of  $p$  being prime cannot be removed.
+[[people:fer:402ws:spring2019:previous_homework|Previous Homework]]
+[[people:fer:402ws:spring2019:home| Home]]

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