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189-570A: Higher Algebra I

Assignment 6 (Practice final)

You do not need to hand in your work, although I will correct it if you wish.
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.

1. Let $G$ be a finite group and let $V$ be a finite-dimensional representation of $G$ over a field $F$ of characteristic $p$, where $p$ is a prime that does not divide the cardinality of $G$.

(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 $V$).

2. Let $G=S_4$ be the permutation group on $4$ elements.

(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 $G$.

3. A finite group $G$ is said to be rational if all the entries in its character table are rational numbers.

(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.

4. Let $F={\bf R}(t)$ be the field of rational functions in an indeterminate $t$ over the field of real numbers, and let $K = {\bf C}(t^{1/p})$, where $p$ is a prime number.

(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 $K$.

5. Let $K/F$ be a finite seperable extension of fields.

(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.

6. Let $G_{\mathbb Q}$ be the absolute Galois group of $\mathbb Q$, i.e., the automorphism group of (an) algebraic closure ${\overline{ \mathbb Q}}$ of $\mathbb Q$, equipped with its Krull topology.

(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 infinite image.

7. (Extra credit question. The actual exam will probably have only 6 questions that need to be written.)

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 $k(t)$.