Problem Set 10 (complete) Due: 05/10/2019

1. Let $F$ be a field of characteristic zero, $a\in F$, and $\xi=\xi_n$ a primitive $n$-th root of unity.
   1. Show by example that $\gal_F(F(\xi))$ need not be all of $U_n$.
   2. Show by example that $\gal_{F(\xi)}(F(\xi,\sq[n]{a}))$ need not be all of $C_n$.
2. Let $G$ and $H$ be solvable groups. Prove that $G\times H$ is solvable.
3. 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

1. 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.
2. Let $\xi_{15}=\cis(2\pi/15)$ be a primitive $15$-th root of unity.
   1. Find the group $\gal(\Q(\xi_{15})/\Q)$ and draw its lattice of subgroups.
   2. Find and draw the lattice of intermediate fields of the extension $\Q(\xi_{15})/\Q$.
   3. Write down the correspondence between the subgroups in part 1, and the subfields in part 2, using the Fundamental Theorem of Galois Theory.
3. Show that any non-abelian simple group is non-solvable.
4. 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

1. 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 $H_1,H_2\in\sub(G)$.
   1. $(L_1\meet L_2)^* = L_1^* \join L_2^*$
   2. $(L_1\join L_2)^* = L_1^* \meet L_2^*$
   3. $(H_1\meet H_2)^* = H_1^* \join H_2^*$
   4. $(H_1\join H_2)^* = H_1^* \meet H_2^*$
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.
3. Consider the polynomial $f(x)=x^3+2x^2+2x+2\in\Q[x]$, and $E$ its splitting field over $\Q$.
   1. Show that $f(x)$ has exactly one real root. (Hint: use calculus)
   2. Show that $f(x)$ is irreducible over $\Q$.
   3. Find $[E:\Q]$. Fully explain your calculation.
   4. Determine $\gal_\Q(E)$.
4. Consider the group $S_n$ of all permutations of the set $\{1,2,\dots,n\}$.
   1. Show that the transpositions $(1\ \ 2),(2\ \ 3),\dots,(n-1\ \ n)$ generate the whole group $S_n$.
   2. 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 $\rho$.)
   3. 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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