PRELIMINARY EXAMINATION IN ALGEBRA

Compute the number of elements of order 4 in the symmetric group \(S_7\).

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

Let \(E\subset F\) be a finite Galois extension of fields. Suppose that there is an element \(\alpha \) in \(F\) such that \(\alpha \notin E\) and such that \(\alpha \) is in every proper extension of \(E\) contained in \(F\). Show that the Galois group of \(F\) over \(E\) is cyclic of prime power order.

Show that every finitely generated subgroup of \(\mathbb {Q}/\mathbb {Z}\) is cyclic.

Show that there are no simple groups of order 520.

Let \(M\) be a matrix with real coefficients such that \(M^2+M+I=0\). Give the possible canonical forms for \(M\)

1. Over the real numbers \(\mathbb {R}\) (rational canonical form).
2. Over the complex numbers \(\mathbb {C}\) (Jordan canonical form).

Let \(R\) be a commutative ring, and let \(S\) be a multiplicative subset of \(R\) (\(1\in S\), and if \(s\) and \(t\) are in \(S\), then \(st\) is in \(S\)). Show that an ideal \(I\) that is maximal with the property that \(S\cap I=\emptyset \) is prime.

Determine for which integers \(q\) the polynomial \(X^3+1\) has three distinct roots in the field with \(q\) elements.

Show that there exists a Galois extension of the field \(\mathbb {Q}\) of rational numbers with Galois group \(\mathbb {Z}/5\mathbb {Z}\).

State and prove the Eisenstein Irreducibility Criterion.