PRELIMINARY EXAMINATION IN ALGEBRA August 13, 2002

Let \(G\) be a group of order \(p^n\), where \(p\) is a prime number and \(n>0\) is an integer. Show that the center of \(G\) is not trivial.

Let \(E\subset F\) be an extension of finite fields. Let \(q\) be the number of elements of \(E\), and let \(\phi \colon F\to F\) be defined by \(\phi (a)=a^q\). Prove that \(F\) is a Galois extension of \(E\) and that \(\phi \) generates its Galois group.

Describe the 2-Sylow subgroup of \(S_6\) (that is, show that it is isomorphic to a group described in terms of known groups).

Prove that a finite abelian group is determined up to isomorphism by the number of elements of each order.

Show that there are no simple groups of order 616.

Let \(E\subset F\) be an extension of finite fields. Prove that the trace map from \(F\) to \(E\) is surjective.

Let \(G\) be a finite group, and let \(C\) be the category of finitely generated \(\mathbb {Q}[G]\)-modules. Show the the functor which takes a \(\mathbb {Q}[G]\)-module to its submodule of invariants (that is, the set of \(m\in M\) such that \(gm=m\) for all \(g\in G\)) is exact.

Determine for which fields the polynomial \(X^3-X+1\) has a multiple root.

Let \(E\) be a finite separable extension of a field \(K\) and let \(F\) be any finite field extension of \(K\). Show that \(E\otimes _K F\) is a finite product of fields.

Prove that a principal ideal domain is a unique factorization domain.

Let \(R\) be a an algebra over a field \(k\) of dimension \(n\). Let \(I\) be the intersection of all maximal left ideals of \(R\). Show that \(I\) is a 2-sided ideal and that \(I^n=0\). Give an example for which \(I^{n-1}\ne 0\).