Algebra Preliminary Exam, Fall 1993

Do all problems

1.
Assume that R is a ring and e $ \in$ R has the property that e2 = e. Prove that Re is a projective left R-module.
2.
Let F be a field of characteristic zero and let K be a finite field extension of F.
(a)
Explain why there is a polynomial p(X) $ \in$ F[X] such that K $ \cong$ F[X]/(p).
(b)
Prove that if c $ \in$ K[X]/(p) and c2 = 0, then c = 0.

3.
Let M 2($ \mathbb {Q} $) denote the group of 2 X 2 matrices with rational entries under addition, and let GL 2($ \mathbb {Q} $) denote the group of invertible 2 X 2 matrices with rational entries under multiplication.
(a)
Prove that if M 2($ \mathbb {Q} $) acts on a set, then all orbits are either infinite or singletons.
(b)
Show that GL 2($ \mathbb {Q} $) acts on $ \mathbb {Q} $ via g*l = det(g)l / | det (g)|, and that there exists a finite orbit which is not a singleton.

4.
Let G be the direct product of the dihedral group of order 34 and the cyclic group of order 9. Suppose that L is a field and G is a group of automorphisms of L. Prove that there is a unique field K such that LG $ \subseteq$ K and dimKL = 17. (You may assume that charL = 0.)

5.
Let S be a commutative integral domain. Prove that if every prime ideal of S[X] is principal, then S is a field.

6.
Let A be an abelian group.
(a)
Show that the collection H of all homomorphisms from A to $ \mathbb {Z} $ is a group under addition of functions.
(b)
Prove that if f1,..., fm $ \in$ H, then the subgroup generated by f1,..., fm is free (i.e. free as a $ \mathbb {Z} $ -module).

7.
Let p be a prime, and let G be the group of invertible 2 X 2 matrices under multiplication with entries in the field of integers modulo p. Let H be the subgroup consisting of all matrices of the form
(
1 0
a 1
)
(a)
Show that | G| = (p2 - 1)(p2 - p).
(b)
Find all values of p such that the number of conjugates of H in G is congruent to 8 mod p.


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Peter Linnell
1999-05-31