## Algebra Prelim 2013

INSTRUCTIONS: Do as many of the eight problems as you can.
Four completely correct solutions will be a pass; a few complete solutions will count more than many partial solutions.
Always carefully justify your answers.
If you skip a step or omit some details in a proof, point out the gap and, if possible, indicate what would be required to fill it in.

Notation: Q is the field of rational numbers and C is the field of complex numbers.

Problem 1. Let $\mathbf { Q } ^ { \times }$ be the nonzero elements of Q, a group under multiplication.

(a) Prove that the additive group of Q has no maximal proper subgroups.

(b) Is the same statement true for the multiplicative group $\mathbf { Q } ^ { \times \ ? }$

Problem 2. Let V be a finite-dimensional vector space over a field F of characteristic 0. Let $B : V \times V  F$ be a non-degenerate, skew-symmetric bilinear form.
(In particular, we have $B ( x , y ) = - B ( y , x )$ for all $x , y \in V . )$ If U is a subset of V , let

$$
U ^ { \perp } \ = \ \{ v \in V \mid B ( u , v ) = 0 { \mathrm { ~ f o r ~ a l l ~ } } u \in U \ \} .
$$

(a) Let U be a subspace of V . Prove that $U ^ { \perp }$ is a subspace of V and that

$$
\dim _ { F } ( U ) + \dim _ { F } ( U ^ { \perp } ) = \dim _ { F } ( V ) \ .
$$

(b) Prove that there exists a subspace W of V such that $W ^ { \perp } = W$

## Problem 3.

(a) Suppose that G is a finitely-generated group.
Let n be a positive integer.
Prove that G has only finitely many subgroups of index n.

(b) Let p be a prime number.
If G is any finitely-generated abelian group, let $t _ { p } ( G )$ denote the number of subgroups of G of index p. Determine the possible values of $t _ { p } ( G )$ as G varies over all finitely-generated abelian groups.

Problem 4. Suppose that G is a finite group of order 2013. Prove that G has a normal subgroup N of index 3 and that N is a cyclic group.
Furthermore, prove that the center of G has order divisible by 11. ( You will need the factorization $2 0 1 3 = 3 \cdot 1 1 \cdot 6 1 . .$ )

Problem 5. Let V be a finite dimensional vector space over C. Let $n = d i m _ { \bf C } ( V )$ . Let $T : V  V$ be a linear map.
Suppose that the following statement is true.

For every $c \in \mathbf { C } ,$ the subspace $\{ v \in V \mid T ( v ) = c v \}$ of V has dimension 0 or 1.

Prove that there exists a vector $w \in V$ such that $\{ w , \ T ( w ) , \ . . . . , \ T ^ { n - 1 } ( w ) \}$ is a linearly independent set.

Problem 6. This question concerns an extension K of Q such that $\left[ K : \mathbf { Q } \right] = 8$ . Assume that $K / \mathbf { Q }$ is Galois and let $G = \operatorname { G a l } ( K / \mathbf { Q } )$ . Furthermore, assume that G is nonabelian.

(a) Prove that K has a unique subfield F such that $F / \mathbf { Q }$ is Galois and $[ F : \mathbf { Q } ] = 4 { \mathrm { . } }$

(b) Prove that F has the form $F = \mathbf { Q } ( { \sqrt { d _ { 1 } } } , { \sqrt { d _ { 2 } } } )$ , where $d _ { 1 }$ and $d _ { 2 }$ are nonzero integers.

(c) Suppose that G is the quaternionic group.
Prove that $d _ { 1 }$ and $d _ { 2 }$ are positive integers.

Problem 7. Let $R = \mathbf { C } [ x _ { 1 } , . . . , x _ { n } ]$ be the polynomial ring over C in n indeterminates $x _ { 1 } , . . . , x _ { n }$ . Let $S _ { n }$ be the n-th symmetric group.
If $\sigma \in S _ { n }$ , then we can identify σ with the automorphism of R defined as follows: $\sigma ( c ) = c$ for all $c \in \mathbf { C }$ , and $\sigma ( x _ { i } ) = x _ { \sigma ( i ) }$ for all i, $1 \leq i \leq n$ . Suppose that G is any subgroup of $S _ { n }$ . Let

$$
S = R ^ { G } = \{ r \in R \mid \sigma ( r ) = r { \mathrm { ~ f o r ~ a l l ~ } } \sigma \in G \}
$$

Prove that S is a finitely-generated C-algebra.

Problem 8. This question concerns the polynomial ring ${ \cal R } = { \bf Z } [ x , y ]$ and the ideal $I = ( 5 , x ^ { 2 } + 2 )$ in R.

(a) Prove that I is a prime ideal of R and that $R / I$ is a PID.

(b) Give an explicit example of a maximal ideal of R which contains I. (Give a set of generators for such an ideal.)

(c) Show that there are infinitely many distinct maximal ideals in R which contain I.