We can discuss any questions about this assignment in office hours or in the last week, which will be devoted to review of the material.

(a) If $W\subset V$ is a $G$-stable vector subspace of $G$, show that it admits a $G$-stable complement, i.e, a $G$-stable subspace $W'\subset V$ for which $V = W\oplus W'$.

(b) Give an example to show that the conclusion of (a) can fail when $p$ divides the cardinality of $G$.

(c) If $G$ is a group of cardinality a power of $p$, and $V$ is isomorphic to the direct sum of irreducible representations of $G$, show that $G$ acts trivially on $V$ (i.e., every element of $G$ acts as the identity transformation of $V$).

(a) List the conjugacy classes in $G$, along with their sizes.

(b) Write down the character table for $G$.

(c) Explain how the character table you have computed can be used to give a complete list of all the normal subgroups of $G$.

(a) Show that $G$ is rational if and only if any two elements that generate the same cyclic subgroup of $G$ are conjugate to each other.

(b) Use the result of (a) to show that the symmetric group $S_n$ is rational, while the group $GL_2({\bf F}_p)$ of invertible matrices with entries in the finite field with $p$ elements is not.

(a) Compute the degree $[K:F]$ and show that $K/F$ is a Galois extension.

(b) Compute the Galois group $G$ of $K/F$.

(c) Use the Galois correspondence to give a complete list of the extensions of $F$ of degree $p$ that are contained in $K$.

(a) Show that the number of field extensions of $F$ contained in $K$ is finite.

(b) Show that this finiteness assertion ceases to hold if the assumption that $K/F$ is seperable is dropped.

(a) Show that there are no continuous homomorphisms from $G_{\mathbb Q}$ into the additive group of ${\mathbb C}$ (equipped with its usual Euclidean topology).

(b) Show that any homomorphism from $G_{\mathbb Q}$ to $\mathbb C^\times$ has for image a finite cyclic group.

(c) (Extra credit)

Show that there are continuous homomorphisms from $G_{\mathbb Q}$ to $\mathbb Q_p^\times$ with

Let $k$ be the finite field with $p$ elements ($p$ a prime), and let Let $K=k(x)$ be the field of rational functions in the variable $x$. Let $G$ be the group of automorphisms of $K$ obtained by the ``mobius transformations" $$x \mapsto \frac{ax + b}{cx+d}, \quad \mbox{with } a,b,c,d \in k, \ \ ad-bc\ne 0.$$ For any extension $k'$ of $k$, this rule also equipes the projective line $\mathbb P_1(k') = k'\cup\{\infty\}$ with an action of $G$.

(a) Show that $G$ acts transitively on $\mathbb P_1(k)$ and that, for any $a\in \mathbb P_1(k)$, $$f_1(x) := \prod (x-g(a)) = (x^p-x)^{p(p-1)},$$ where the product is taken over all $g\in G$ for which $g(a)\ne \infty$.

(b) Let $k_2$ be the unique quadratic extension of $k$. Show that $G$ acts transitively on $\mathbb P_1(k_2) - \mathbb P_1(k) = k_2-k$ and that, for all $b\in k_2-k$, $$ f_2(x) := \prod (x-g(b)) = \left(\frac{x^{p^2}-x}{x^p-x}\right)^{p+1},$$ where the product is taken over all $g\in G$.

(c) Let $k_3$ be the unique cubic extension of $k$. Show that $G$ acts transitively on $\mathbb P_1(k_3) - \mathbb P_1(k) = k_3-k$ and that, for all $c\in k_3-k$, $$ f_3(x) := \prod (x-g(c)) = \frac{x^{p^3}-x}{x^p-x},$$ where the product is taken over all $g\in G$.

(d) Show that the rational function $t := f_2(x)/f_1(x)$ generates the field $K^G$ of $G$-invariant elements of $K$, and likewise for $f_3(x)/f_1(x)$. Conclude that the polynomials $$ f_2(x) - t f_1(x), \quad f_3(x) - t f_1(x) \in k(t)[x]$$ have Galois group $G$ over $k(t)$.