# 2010 Algebra Prelim

August 31, 2010

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. Let $p$ be a positive prime number, $\mathbb { F } _ { p }$ the field with p elements, and let $G = \mathrm { G L } _ { 2 } ( \mathbb { F } _ { p } )$

(a) Compute the order of $G , | G |$

(b) Write down an explicit isomorphism from $\mathbb { Z } / p \mathbb { Z }$ to

$$
U = \left\{ { \left( \begin{array} { l l } { 1 } & { a } \\ { 0 } & { 1 } \end{array} \right) } \ \left| \ a \in \mathbb { F } _ { p } \right. \right\} .
$$

(c) How many subgroups of order p does G have?
Hint: compute gug−1 for $g \in G$ and $u \in U ;$ use this to find the size of the normalizer of U in $G .$

2. (a) Give definitions of the following terms: (i) a finite length (left) module, (ii) a composition series for a module, and (iii) the length of a module,

(b) Let l(M ) denote the length of a module M . Prove that if

$$
0  M _ { 1 }  M _ { 2 }  \cdot \cdot \cdot  M _ { n }  0
$$

is an exact sequence of modules of finite length, then

$$
\sum _ { i = 1 } ^ { n } ( - 1 ) ^ { i } l ( M _ { i } ) = 0 .
$$

3. Let $\mathbb { F }$ be a field of characteristic $p ,$ and G a group of order $p ^ { n }$ . Let $R = \mathbb { F } [ G ]$ be the group ring (group algebra) of G over F, and let $\textstyle u : = \sum _ { x \in G } x$ (so u is an element of R).

(a) Prove that u lies in the center of R.

(b) Verify that Ru is a 2-sided ideal of R.

(c) Show there exists a positive integer k such that $u ^ { k } = 0$ . Conclude that for such a $k , ( R u ) ^ { k } = 0$

(d) Show that R is not a semi-simple ring.
(Warning: Please use the definition of a semisimple ring; do not use the result that a finite length ring fails to be semisimple if and only if it has a non-zero nilpotent ideal.)

4. Let $f ( x ) = a _ { n } x ^ { n } + a _ { n - 1 } x ^ { n - 1 } + \cdots + a _ { 0 } \in \mathbb { Z } [ x ]$ (where $a _ { n } \neq 0 )$ and let $R = \mathbb { Z } [ x ] / ( f )$ Prove that R is a finitely-generated module over Z if and only if $a _ { n } = \pm 1$

5. Consider the ring

$$
S = C [ 0 , 1 ] = \{ f : [ 0 , 1 ] \to \mathbb { R } | f { \mathrm { ~ i s ~ c o n t i n u o u s } } \}
$$

with the usual operations of addition and multiplication of functions.

(a) What are the invertible elements of S?

(b) For $a \in [ 0 , 1 ]$ , define $I _ { a } = \{ f \in S | f ( a ) = 0 \}$ . Show that $I _ { a }$ is a maximal ideal of S .

(c) Show that the elements of any proper ideal of S have a common zero, i.e., if I is a proper ideal of S, then there exists $a \in [ 0 , 1 ]$ such that $f ( a ) = 0$ for all $f \in I$ Conclude that every maximal ideal of S is of the form $I _ { a }$ for some $a \in [ 0 , 1 ]$ Hint: as [0, 1] is compact, every open cover of [0, 1] contains a finite subcover.

6. (a) Let $L / F$ be a field extension that is finite and Galois.
   Show that if the Galois group $\operatorname { G a l } ( L / F )$ is abelian then for every intermediate field $F \subseteq K \subseteq L , K / F$ is also a Galois extension.

(b) Let $K = \mathbb { Q } { \Big ( } { \sqrt { 1 + { \sqrt { 2 } } } } { \Big ) } \subset \mathbb { R }$ . Show that $K / \mathbb { Q }$ is an extension of degree 4 that is not Galois.

(c) Let L be the smallest Galois extension of Q that contains $K = \mathbb { Q } { \Big ( } { \sqrt { 1 + { \sqrt { 2 } } } } { \Big ) }$ Compute the group $\operatorname { G a l } ( L / \mathbb { Q } )$

7. Let F be a field of characteristic zero, and let K be an algebraic extension of F that possesses the following property: every polynomial $f \in F [ x ]$ has a root in K. Show that K is algebraically closed.

Hint: if $K ( \theta ) / K$ is algebraic, consider $F ( \theta ) / F$ and its normal closure; primitive elements might be of help.

8. Let G be the unique non-abelian group of order 21.

(a) Describe all 1-dimensional complex representations of G.

(b) How many (non-isomorphic) irreducible complex representations does G have and what are their dimensions?

(c) Determine the character table of G.