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===== Math 402 - 01 Homework (Spring 2019)=====
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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. 

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