# 2014 Algebra Prelim

September 8, 2014

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 several partial solutions.
Always 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. (a) Let G be a group (not necessarily finite) that contains a subgroup of index n. Show that G contains a normal subgroup N such that $n \leq [ G : N ] \leq n !$

(b) Use part (a) to show that there is no simple group of order 36.

2. Let p be a prime, let $\mathbb { F } _ { p }$ be the p-element field, and let $K = \mathbb { F } _ { p } ( t )$ be the field of rational functions in t with coefficients in $\mathbb { F } _ { p }$ . Consider the polynomial $f ( X ) = X ^ { p } - t \in K [ X ]$

(a) Show that f does not have a root in K.

(b) Let E be the splitting field of f over K. Find the factorization of f over E.

(c) Conclude that f is irreducible over K.

3. Recall that a ring A is called graded if it admits a direct sum decomposition $A =$ $\textstyle \bigoplus _ { n = 0 } ^ { \infty } A _ { n }$ as abelian groups, with the property that $A _ { i } A _ { j } \subseteq A _ { i + j }$ for all $i , j \geq 0$ Prove that a graded commutative ring $A = \oplus _ { n = 0 } ^ { \infty } A _ { n }$ is Noetherian if and only if $A _ { 0 }$ is Noetherian and A is finitely generated as an algebra over $A _ { 0 }$

4. Let R be a ring with the property that $a ^ { 2 } = a$ for all $a \in R$

(a) Compute the Jacobson radical of R.

(b) What is the characteristic of R?

(c) Prove that R is commutative.

(d) Prove that if R is finite, then R is isomorphic (as a ring) to $( \mathbb { Z } / 2 \mathbb { Z } ) ^ { d }$ for some d.

5. Let R be a commutative ring and let M be an R-module.

(a) Let $x \in R$ be a nonzero divisor.
Compute $\mathrm { T o r } _ { i } ^ { R } ( R / ( x ) , M )$ for $i \geq 0$ (b) Show that M is a flat R-module if and only if $\mathrm { T o r } _ { 1 } ^ { R } ( M , N ) = 0$ for all R-modules N .

(c) Conclude that if R is a PID and M is a finitely generated R-module, then M is flat if and only if M is free.

6. Let $\overline { { \mathbb { F } _ { p } } }$ denote the algebraic closure of $\mathbb { F } _ { p }$ . Show that the Galois group $\operatorname { G a l } ( { \overline { { \mathbb { F } _ { p } } } } / \mathbb { F } _ { p } )$ has no nontrivial finite subgroups.

7. Let $C _ { p }$ denote the cyclic group of prime order $p .$ .

(a) Show that $C _ { p }$ has two irreducible representations over Q (up to isomorphism), one of dimension 1 and one of dimension $p - 1$

(b) Let G be a finite group, and let $\rho : G \longrightarrow \mathrm { G L } _ { n } ( \mathbb { Q } )$ be a representation of G over Q. Let $\rho _ { \mathbb { C } } : G \longrightarrow { \mathrm { G L } } _ { n } ( \mathbb { C } )$ denote ρ followed by the inclusion ${ \mathrm { G L } } _ { n } ( \mathbb { Q } ) \longrightarrow$ ${ \mathrm { G L } } _ { n } ( \mathbb { C } )$ . Thus $\rho _ { \mathbb { C } }$ is a representation of G over C, called the complexification of $\rho .$ We say that an irreducible representation $\rho$ of G is absolutely irreducible if its complexification remains irreducible over C.

Now suppose G is abelian and that every representation of G over Q is absolutely irreducible.
Show that $G \cong ( C _ { 2 } ) ^ { k }$ for some k (i.e., is a product of cyclic groups of order 2).

8. Let G be a finite group and $\mathbb { Z } [ G ]$ the integral group algebra.
   Let $\mathcal { Z }$ be the center of Z[G]. For each conjugacy class $C \subset G$ , let $\textstyle P _ { C } = \sum _ { g \in C } g$

(a) Show that the elements $P _ { C }$ form a Z-basis for $\mathcal { Z }$ . Hence $\mathcal { Z } \cong \mathbb { Z } ^ { d }$ as an abelian group, where d is the number of conjugacy classes in G.

(b) Show that if a ring R is isomorphic to $\mathbb { Z } ^ { d }$ as an abelian group, then every element in R satisfies a monic integral polynomial.
(Hint: Let $\{ v _ { 1 } , \ldots , v _ { d } \}$ be a basis of R and for a fixed non-zero $r \in R$ , write $\begin{array} { r } { r v _ { i } = \sum _ { j } a _ { i j } v _ { j } } \end{array}$ . Use the Hamilton-Cayley theorem.)

(c) Let $\pi : G \longrightarrow \mathrm { G L } ( V )$ be an irreducible representation of G (over C). Show that $\pi ( P _ { C } )$ acts on V as multiplication by the scalar

$$
{ \frac { | C | \chi _ { \pi } ( C ) } { \dim V } } ,
$$

where $\chi _ { \pi } ( C )$ is the value of the character $\chi _ { \pi }$ on any element of C.

(d) Conclude that $| C | \chi _ { \pi } ( C ) /$ dim V is an algebraic integer.