# Algebra Prelim September 12, 2011

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.

1. Let $\mathrm { G L _ { 2 } ( \mathbb { C } ) }$ be the general linear group of $2 \times 2$ complex matrices, let H be the subgroup of $\mathrm { G L _ { 2 } ( \mathbb { C } ) }$ consisting of non-zero multiples of the identity matrix, and let $\mathrm { P G L _ { 2 } ( \mathbb { C } ) }$ b e the quotient group $\mathrm { G L _ { 2 } ( C ) / H }$

Let $A , B \in \mathrm { P G L _ { 2 } } ( \mathbb { C } )$ , and assume that both elements have order n. Prove that there exist $C \in \mathrm { P G L } _ { 2 } ( \mathbb { C } )$ and a positive integer m such that

$$
C B C ^ { - 1 } = A ^ { m } .
$$

2. In this problem, as you apply Sylow’s Theorem, state precisely which portions you are using.

(a) Prove that there is no simple group of order 30.

(b) Suppose that G is a simple group of order 60. Determine the number of p-Sylow subgroups of G for each prime p dividing 60, then prove that G is isomorphic to the alternating group $A _ { 5 }$

Note: In the second part, you needn’t show that $A _ { 5 }$ is simple.
You need only show that if there is a simple group of order 60, then it must be isomorphic to $A _ { 5 }$

3. Describe the Galois group and the intermediate fields of the cyclotomic extension $\mathbb { Q } ( \zeta _ { 1 2 } ) / \mathbb { Q }$

4. Let

$$
R = \mathbb { Z } [ x ] / ( x ^ { 2 } + x + 1 ) .
$$

(a) Answer the following questions with suitable justification.

i. Is R a Noetherian ring?

ii.
Is R an Artinian ring?

(b) Prove that R is an integrally closed domain.

5. Let R be a commutative ring.
   Recall that an element r of R is nilpotent if $r ^ { n } = 0$ for some positive integer n and that the nilradical of R is the set $N ( R )$ of nilpotent elements.

(a) Prove that

$$
N ( R ) = \bigcap _ { P { \mathrm { ~ p r i m e } } } P .
$$

(Hint: Given a non-nilpotent element r of R, you may wish to construct a prime ideal that does not contain r or its powers.)

(b) Given a positive integer m, determine the nilradical of $\mathbb { Z } / ( m )$

(c) Determine the nilradical of $\mathbb { C } [ x , y ] / ( y ^ { 2 } - x ^ { 3 } )$

(d) Let $p ( x , y )$ be a polynomial in $\mathbb { C } [ x , y ]$ such that for any complex number a, $p ( a , a ^ { 3 / 2 } ) = 0$ . Prove that $p ( x , y )$ is divisible by $y ^ { 2 } - x ^ { 3 }$

6. Given a finite group G, recall that its regular representation is the representation on the complex group algebra $\mathbb { C } [ G ]$ induced by left multiplication of G on itself and its adjoint representation is the representation on the complex group algebra C[G] induced by conjugation of G on itself.

(a) Let $G = \mathrm { { G L } _ { 2 } ( \mathbb { F } _ { 2 } ) }$ . Describe the number and dimensions of the irreducible representations of G. Then describe the decomposition of its regular representation as a direct sum of irreducible representations.

(b) Let H be a group of order 12. Show that its adjoint representation is reducible; that is, there is an H-invariant subspace of $\mathbb { C } [ H ]$ besides 0 and $\mathbb { C } [ H ]$

7. Let M, N be finitely generated modules over Z. Recall that $\mathrm { A n n } ( M )$ is the ideal in Z defined as follows:

$$
\operatorname { A n n } ( M ) = \{ a \in \mathbb { Z } | a m = 0 { \mathrm { ~ f o r ~ a n y ~ } } m \in M \}
$$

Prove that $M \otimes _ { \mathbb { Z } } N = 0$ if and only if $\mathrm { A n n } ( M ) + \mathrm { A n n } ( N ) = ( 1 )$

8. Let R be a commutative integral domain.
   Show that the following are equivalent:

(a) R is a field;

(b) R is a semi-simple ring;

(c) Any R-module is projective.