PRELIMINARY EXAMINATION IN ALGEBRA August 16, 1999

Show that there are no simple groups of order 1365.

Prove that the symmetric group \(S_n\) on \(n\) elements is generated by the transpositions \((12),(23),\ldots ,(n-1\ n)\).

Let \(G\) be a group of order 55 acting on a set \(X\) consisting of 24 elements. Show that there is at least one element fixed by every element of \(G\).

Let \(F\) be a field, and let \(f(X)\) be a polynomial in \(F[X]\). Show that \(f(X)\) has a multiple root in some extension of \(F\) if and only if \(f(X)\) and its derivative \(f'(X)\) are not relatively prime.

Show that if \(\alpha \) and \(\beta \) are elements of a field \(E\) which are algebraic over a subfield \(F\), then \(\alpha +\beta \) and \(\alpha \beta \) are algebraic over \(F\).

Let \(E\) be a normal extension of \(\mathbb {Q}\) contained in \(\overline {\mathbb {Q}}\) (the algebraic closure of \(\mathbb {Q}\)), and let \(K\) be an extension of \(\mathbb {Q}\) contained in \(E\). Let \(K'\) be a subfield of \(\overline {\mathbb {Q}}\) which is isomorphic to \(K\). Show that \(K'\) is contained in \(E\) and the Galois groups of \(E/K\) and \(E/K'\) are conjugate in the Galois group of \(E/\mathbb {Q}\).

Prove that if \(A\) is a commutative Noetherian ring, then \(A[X_1,\ldots ,X_n]\) (the polynomial ring in \(n\) variables over \(A\)) is Noetherian.

Let \(G\) be a finite group. Prove that every short exact sequence of \(\mathbb {C}[G]\)-modules splits, where \(\mathbb {C}[G]\) is the group algebra of \(G\) over the complex numbers.

Let \(K\) be an arbitrary field, and let \(L\) be an extension of \(K\) obtained by adjoining a root of the polynomial \(X^n-1\) for some integer \(n\ge 1\). Show that \(L\) is Galois over \(K\) with abelian Galois group.

Show that a free group \(F\) is torsion-free; that is, that if \(g\ne 1\) is an element of \(F\), then \(g^n\ne 1\) for all \(n\ge 1\).

Let \(M\) be an \(n\) by \(n\) matrix with entries in \(\mathbb {C}\). Assume that there is a polynomial \(f(X)\) in \(\mathbb {Q}[X]\) which is irreducible over \(\mathbb {Q}\) such that \(f(M)=0\) (i.e. it is the zero matrix). Show that \(M\) is diagonalizable over \(\mathbb {C}\).

Show that the endomorphism ring of a simple module is a division ring.