189-571B: Higher Algebra II

To be presented: Friday, March 9.

The problem marked with a (*) is for everyone to do and to hand in on the day of the problem session. I will return this problem to you on Friday of the same week, so late problems will not be accepted!

If you are stuck on your problem, come see me during my Monday office hours (or make an appointment) to get a hint...

1. (*) Let E be a cyclic field extension of F of degree n, and let t:E --> E be a generator of its Galoia group. Let r be an element of F. Let A be the set of all expressions of the form

x0 + x1 u + x2 u2 + ... + xn-1 un-1

where the xi are in E and u is an indeterminate. Make A into an algebra by defining addition in the obvious way and defining the multiplication by the relations

ux = (tx) u, for all x in E.

un = r.

The algebra thus obtained is called the cyclic algebra attached to (E,t,r).

a) Show that A is a central simple algebra over F of degree n (i.e., of dimension n2 over F).

b) (Extra credit) Under what condition on r and E is A a division algebra? You may suppose in this question, to simplify your analysis, that n is prime, so that E has no non-trivial subfields containing F.

c) (Extra credit) For each prime p, write down a division algebra of rank p2 over Q, the field of rational numbers.

2. (Alexandru Stanculescu) If G is a finite group and H is a proper subgroup, show that G is not equal to the union of the conjugates of H. Let D be a finite division algebra with center F. Show that any two maximal subfields E1 and E2 of D are conjugate in D, i.e., there exists a in D such that E1 = a E2 a-1. Conclude that there are no finite non commutative division algebras. (Wedderburn's theorem).

(Christian Cote) 3. Let A be the ring of "Hamilton quaternions over Q". Find necesary and sufficient conditions on a quadratic extension E of Q to be a maximal subfield of A. Write A as a cyclic algebra, as in question 1.