# ALGEBRA PRELIMINARY EXAM – AUTUMN 2016

Do as many of the eight problems as you can.
Four completely correct solutions will be a pass; a few complete solutions will count for 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 G be a finite simple group (that is, a group with no proper nontrivial normal subgroup).
   Assume that every proper subgroup of G is abelian.
   Prove that then G is cyclic of prime order.

2. Let $a \in \mathbb { N } , a > 0$ . Compute the Galois group of the splitting field of the polynomial $x ^ { 5 } - 5 a ^ { 4 } x + a$ over $\mathbb { Q } .$

3. Let ${ \mathfrak { m } } \subset \mathbb { Z } [ x _ { 1 } , \dotsc , x _ { n } ]$ be a maximal ideal.
   Show that $\mathbb { Z } [ x _ { 1 } , \ldots , x _ { n } ] / { \mathfrak { m } }$ is a finite field.

4. Recall that an inner automorphism of a group is an automorphism given by conjugation by an element of the group.
   An outer automorphism is an automorphism that is not inner.

(a) Prove that $S _ { 5 }$ has a subgroup of order 20.

(b) Use the subgroup from (a) to construct a degree 6 permutation representation of $S _ { 5 }$ (i.e., an embedding $S _ { 5 } \hookrightarrow S _ { 6 }$ as a transitive permutation group on 6 letters).

(c) Conclude that $S _ { 6 }$ has an outer automorphism.

5. Let A be a commutative ring and M a finitely generated A-module.
   Define

$$
\mathrm { A n n } ( M ) = \{ a \in A | a m = 0 { \mathrm { ~ f o r ~ a l l ~ } } m \in M \} .
$$

Show that for a prime ideal ${ \mathfrak { p } } \subset A$ , the following are equivalent:

(a) Ann $M \not \subset { \mathfrak { p } }$

(b) The localization of M at the prime ideal p is 0.

(c) M $\otimes _ { A } k ( { \mathfrak { p } } ) = 0$ , where $k ( { \mathfrak { p } } ) = A _ { \mathfrak { p } } / { \mathfrak { p } } A _ { \mathfrak { p } }$ is the residue field of A at p.

6. Let $A = \mathbb { C } [ x , y ] / ( y ^ { 2 } - ( x - 1 ) ^ { 3 } - ( x - 1 ) ^ { 2 } )$

(a) Show that A is an integral domain and sketch the R-points of Spec A.

(b) Find the integral closure of A. Recall that for an integral domain A with a fraction field K, the integral closure of A in K is the set of all elements of K integral over A.

7. Let $R = k [ x , y ]$ where k is a field, and let $I = ( x , y ) R$

(a) Show that

$$
0 \longrightarrow R { \stackrel { \phi } { \longrightarrow } } \Re \oplus R { \stackrel { \psi } { \longrightarrow } } \Re { \longrightarrow } k \qquad \longrightarrow 0
$$

where $\phi ( a ) = ( - y a , x a ) , \psi ( ( a , b ) ) = x a + y b$ for $a , b \in R$ , is a projective resolution of the R-module $k \simeq R / I$

(b) Show that I is not a flat R-module by computing $\mathrm { T o r } _ { i } ^ { R } ( I , k )$

8. Let k be a field of positive characteristic $p , \mathbb { Z } / p$ a cyclic group of order $p ,$ and $k \mathbb { Z } / p$ the group algebra of $\mathbb { Z } / p$ over k.

(a) Let σ be a generator of $\mathbb { Z } / p$ and let $t = \sigma - 1$ . Show that there is an isomorphism $k \mathbb { Z } / p \simeq k [ t ] / t ^ { p }$

(b) Let M be a finite dimensional projective $k \mathbb { Z } / p .$ -module and Σ the linear operator on M induced by the action of σ. Show that

(i) di $\mathrm { m } _ { k } M$ is divisible by $p ;$ and

(ii) $\operatorname { r k } ( \Sigma - \operatorname { I d } _ { M } ) = { \frac { p - 1 } { p } } \dim _ { k } M .$

(c) Let M be a finite dimensional $k \mathbb { Z } / p$ -module, and assume that

$$
\mathrm { r k } ( \Sigma - \mathrm { I d } _ { M } ) = \frac { p - 1 } { p } \dim _ { k } M .
$$

Prove that then M is projective.