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\begin{document}

\begin{center}
{\bf Math 100C: Spring 2017 \\ Homework 7 \\ Due 6/7/2017}
\end{center}





\vspace{0.1in}
\noindent
{\bf Problem 1:}  
Let $F = \QQ$ and let $K = \QQ(\sqrt{2}, \sqrt{3}, \sqrt{5})$.  Determine all intermediate 
fields in the extension $K/F$.

\vspace{0.1in}
\noindent
{\bf Problem 2:}  
Let $K/F$ be a Galois extension with $G(K/F) \cong C_2 \times C_{12}$.  How many 
intermediate extensions $F \subseteq L \subseteq K$ are there with 
(a) $[L:F] = 4$  (b) $[L:F] = 9$ (c) $G(K/L) \cong C_4$?

\vspace{0.1in}
\noindent
{\bf Problem 3:} 
Let $\zeta = \exp(2 \pi i/11)$, let $F = \QQ$, and let $K = \QQ(\zeta)$.  Find all intermediate
fields in the extension $K/F$.



\vspace{0.1in}
\noindent
{\bf Problem 4:}  
Determine the Galois groups of the polynomials $f(x) = x^{12} - 1, g(x) = x^9 - 1 \in \QQ[x]$.

\vspace{0.1in}
\noindent
{\bf Problem 5:}
Is every Galois extension of degree 10 solvable?

\vspace{0.1in}
\noindent
{\bf Problem 6:}
Find a polynomial $f(x) \in \QQ[x]$ of degree $7$ whose Galois group is $S_7$.

\vspace{0.1in}
\noindent
{\bf Problem 7:}
Let $F = \QQ$ and let $K/F$ be a Galois extension whose Galois group $G(K/F)$ is isomorphic to the dihedral group
$$
G(K/F) \cong D_n = \langle \sigma, \tau \mid \sigma^n = \tau^2 = 1, \, \, \tau \sigma \tau^{-1} = \sigma^{-1} \rangle.
$$
Prove that $K$ is the splitting field over $F$ of an irreducible polynomial $f(x) \in F[x]$ of degree $n$.
(Hint: Let $L = K^{\langle \tau \rangle}$ and let $\alpha \in L$ be a primitive element for the extension
$L/F$.  Try the polynomial $f(x) = (x - \alpha)(x - \sigma(\alpha)) \cdots (x - \sigma^{n-1}(\alpha))$.)



\vspace{0.1in}
\noindent
{\bf Problem 8:}  
{\bf Galois Theory for Finite Fields.}
Let $K/F$ be a (finite) extension of finite fields with $[K:F] = n$.
Prove that the Galois group
$$
G = G(K/F) = \{ \sigma \in \mathrm{Aut}(K) \,:\, \sigma(\alpha) = \alpha \text{ for all $\alpha \in F$} \}
$$
is cyclic of order $n$.  Prove that we have mutually inverse bijections
$$
\{ \text{intermediate extensions $F \subseteq L \subseteq K$} \} \leftrightarrow
\{ \text{subgroups $H \subseteq G$} \}
$$
given by $L \leadsto G(K/L)$ and $H \leadsto K^H$.


\vspace{0.1in}
\noindent
{\bf Problem 9:} 
(Optional - not to be handed in.)
{\bf Induction with Tensor Products.}
Let $G$ be a finite group and let $H$ be a subgroup of $G$.  If $V$ is a $\CC[H]$-module, the {\em induction}
of $V$ from $G$ to $H$ is
$$
\mathrm{Ind}_H^G(V) := \CC[G] \otimes_{\CC[H]} V,
$$
which is a $\CC[G]$-module.

(a)  Let $\{t_1, \dots, t_r\}$ be a left transversal for $H$ in $G$.  Let $\{v_1, \dots, v_n\}$ be a $\CC$-basis of $V$.  Show that
$\{ t_i \otimes v_j \,:\, 1 \leq i \leq r, 1 \leq j \leq n\}$ is a $\CC$-basis for $\mathrm{Ind}_H^G(V)$.

(b)  Prove that the representing matrices for elements of $G$ with respect to the $\CC$-basis from Part (a) are precisely the
`block matrix' version of induction from a previous homework set.

(c)   Let $W$ be a $\CC[G]$-module.
Use the Universal Property of the Tensor Product to produce a $\CC$-vector space isomorphism:
$$
\mathrm{Hom}_{\CC[G]}( \mathrm{Ind}_H^G(V), W) \cong \mathrm{Hom}_{\CC[H]} (V, \mathrm{Hom}_{\CC[G]}(\CC[G],W)).
$$




\vspace{0.1in}
\noindent
{\bf Problem 10:} 
(Optional - not to be handed in.)
{\bf Representations of $GL_n$.}
Consider the group $G = GL_n(\CC)$ of $n \times n$ invertible complex matrices.
Let $\sigma \in \mathrm{Aut}(\CC)$ be any field automorphism of the complex numbers.
\footnote{There are {\em many} of these.}  Prove that the map
$$
R: G \rightarrow G
$$
defined by $R: (a_{i,j}) \mapsto (\sigma(a_{i,j}))$ is a matrix representation of $G$.  When $n = 1$,
prove that any two such representations are non-isomorphic.  This shows that classifying general representations
of $G$ is essentially {\em hopeless}.

\vspace{0.1in}
\noindent
{\bf Problem 11:} 
(Optional - not to be handed in.)
{\bf Weyl Characters.}
Now let $G = GL_n(\CC)$ and let $R: G \rightarrow GL_m(\CC)$ be a {\em continuous}
representation of $G$.  The {\em character} $\chi_R: G \rightarrow \CC$ of $R$ is defined as before
$$
\chi_R(A) := \mathrm{trace}(R_A).
$$
The {\em Weyl character} is the function $\psi_R: (\CC^{\times})^n \rightarrow \CC$ defined by
$$
\psi_R(x_1, \dots, x_n) = \chi_R(\mathrm{diag}(x_1, \dots, x_n)).
$$

(a)  Prove that isomorphic $G$-modules have the same character.

(b)  Prove that the character $\chi_R$ is determined by the Weyl character $\psi_R$. (Hint: Diagonalizable matrices
are dense in $G$.)

(c)  Consider the $G$-module $V = \CC^n$, with the natural action $A.v := Av$ (matrix multiplication).  Prove that the
Weyl character is given by
$$
\psi_V(x_1, \dots, x_n) = x_1 + \cdots + x_n.
$$

(d)  Let $0 \leq k \leq n$.  Let $V$ be the {\em exterior power} $V = \wedge^k \CC^n$.  That is, $V$ is the 
$k$-fold tensor product $\CC^n \otimes \cdots \otimes \CC^n$ modulo elements of the form
$$
v_1 \otimes \cdots \otimes v_i \otimes v_{i+1} \otimes \cdots \otimes v_k +
v_1 \otimes \cdots \otimes v_{i+1} \otimes v_i \otimes \cdots \otimes v_k.
$$
The image of $v_1 \otimes \cdots \otimes v_k$ in $V$ is written $v_1 \wedge \cdots \wedge v_k$.  Then $V$ is a $G$-module
via the rule
$$
A.(v_1 \wedge \cdots \wedge v_k) := (A v_1) \wedge \cdots \wedge (A v_k).
$$
Prove that the Weyl character of $V$ is
$$
\psi_V(x_1, \dots, x_n) = s_k(x_1, \dots, x_n) = \sum_{1 \leq i_1 < \cdots < i_k \leq n} x_{i_1} \cdots x_{i_k},
$$
the degree $k$ elementary symmetric function.

(e)  Let $k \geq 0$.  Let $V$ be the {\em symmetric power} $V = \mathrm{Sym}^k(\CC^n)$.
That is, $V$ is the 
$k$-fold tensor product $\CC^n \otimes \cdots \otimes \CC^n$ modulo elements of the form
$$
v_1 \otimes \cdots \otimes v_i \otimes v_{i+1} \otimes \cdots \otimes v_k -
v_1 \otimes \cdots \otimes v_{i+1} \otimes v_i \otimes \cdots \otimes v_k.
$$
The image of $v_1 \otimes \cdots \otimes v_k$ in $V$ is written $v_1 \cdot \cdots \cdot v_k$.  Then $V$ is a $G$-module
via the rule
$$
A.(v_1 \cdot \cdots \cdot v_k) := (A v_1) \cdot \cdots \cdot (A v_k).
$$
Prove that the Weyl character of $V$ is 
$$
\psi_V(x_1, \dots, x_n) = \sum_{1 \leq i_1 \leq \cdots \leq i_k \leq n} x_{i_1} \cdots x_{i_k}.
$$

(f)  Let $V$ be {\em any} continuous representation of $G$.  Prove that the Weyl character
$\psi_V(x_1, \dots, x_n)$ is a symmetric function of $x_1, \dots, x_n$.


\end{document}








