ALGEBRA PRELIM September, 1995

If \(A\) is a square matrix over the field of complex numbers, prove that \(A\) is similar to its transpose.

Determine, with proof, all rings \(R\) such that \(R\) contains at most 10 elements and \(x^{10}=x\) for all \(x\in R\).

Suppose \(G\) is a finite simple group such that \(G\) has exactly 5 distinct Sylow \(p\)-subgroups for some prime \(p\). Prove that \(G\) is isomorphic to \(A_5\), the alternating group of order 60.

Suppose \(F\) is a field of characteristic 0 with the property that every polynomial of odd degree over \(F\) has at least one root in \(F\). Assume that \(E\) is an extension of degree 2 over \(F\) with the property that for each \(\alpha \in E\), the polynomial \(x^2-\alpha \) has a root in \(E\). Prove that \(E\) is algebraically closed.

If \(R\) is a simple ring and \(n\) is a positive integer, prove that \(R_n\), the ring of all \(n\times n\) matrices over \(R\), is simple.

1. Let \(H\) be a maximal subgroup of the finite group \(G\) and let \(M\) be a minimal normal subgroup of \(G\). Assume that \(M\) is abelian but \(H\) does not contain \(M\). Prove that \(G=HM\), \(H\cap M=\{1\}\) (the identity subgroup), and \(|G:H|\) (the index of \(H\) in \(G\)) is a power of a prime.
2. Use the result in (a) to show that that if \(G\) is a finite solvable group and \(H\) is a maximal subgroup of \(G\), then \(|G:H|\) is a power of a prime.

Let \(F\) be the field of integers modulo 7. Determine which finite extensions of \(F\) contain exactly four 4th roots of 1.

Let \(R\) be a commutative ring with an identity and assume that \(I\) and \(J\) are ideals in \(R\) such that \(I+J=R\). Prove that \(IJ=I\cap J\) and that \(R/IJ\) is isomorphic to the direct product \((R/I)\times (R/J)\).

Assume that \(A\) and \(B\) are solvable subgroups of the group \(G\) and \(G=AB\). Here \(AB=\{ab\mid a\in A,\ b\in B\}\).

1. Give an example showing that \(G\) does not have to be solvable.
2. Prove that if either \(A\) or \(B\) is a normal subgroup of \(G\), then \(G\) is solvable.

Let \(R\) be a ring and let \(M\) be an irreducible \(R\)-module. Prove that one of the following statements is true about \(M\) considered as an abelian group:

1. \(M\) is the direct sum of (perhaps infinitely many) copies of the additive group of all rational numbers.
2. \(M\) is the direct sum of (perhaps infinitely many) copies of a cyclic group of order \(p\) for some prime \(p\).