# 2018 Algebra Prelim

September 10, 2018

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.
Please start each solution on a new page and submit your solutions in order.

1. Show that no finite group is the union of conjugates of a proper subgroup.

2. Classify all groups of order 18 up to isomorphism.

3. Let $\alpha , \beta$ denote the unique positive real 5th root of 7 and 4th root of 5, respectively.
   Determine the degree of $\mathbb { Q } [ \alpha , \beta ]$ over $\mathbb { Q } .$

4. Show that the field extension $\mathbb { Q } \subset \mathbb { Q } [ { \sqrt { 2 + { \sqrt { 2 } } } } ]$ is Galois and determine its Galois group.

5. Let M be a square matrix over a field K. Use a suitable canonical form to show that M is similar to its transpose $M ^ { T }$

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

(a) Show that $M = 0$ if and only if the localization $M _ { \mathfrak { m } } = 0$ for all maximal ideals m of R.

(b) Find an example of a local ring R with maximal ideal m and a nonzero R-module M such that $M / { \mathfrak { m } } M = 0$

7. Let G be a finite group and $\pi , \pi ^ { \prime }$ be two irreducible representations of G. Prove or disprove the following assertion: π and $\pi ^ { \prime }$ are equivalent if and only if det $\pi ( g ) = \operatorname* { d e t } \pi ^ { \prime } ( g )$ for all $g \in G$

8. Let K be a field and $R = K [ x ] / ( x ^ { 2 } )$ . For each integer $i \geq 0$ , compute

(a) $\mathrm { E x t } _ { R } ^ { i } ( R , R / ( x ) )$

(b) $\mathrm { E x t } _ { R } ^ { i } ( R / ( x ) , R )$

(c) $\mathrm { E x t } _ { R } ^ { i } ( R / ( x ) , R / ( x ) )$