ALGEBRA PRELIM EXAM January, 2007

Let \(F\) be a field. Let \(M_5(F)\) denote the set of 5-by-5 matrices with entries in \(F\), and let \(\mathcal {N}\) denote the subset of nilpotent matrices, \[ \mathcal {N}:=\{A\in M_5(F)\mid \text {there exists }N\text { such that }A^N=0\}. \] Let \(X\) denote the subset of \(M_5(F)\) consisting of those matrices which are squares of nilpotent matrices, \[ X:=\{M\in M_5(F)\mid \text {there exists }A\in \mathcal {N}\text { such that }M=A^2\}. \] Observe that \(G=\operatorname {GL}(5,F)\) acts on \(X\) by conjugation. Determine, with proof, the number of orbits for the action of \(G\) on \(X\).

Let \(R\) be a commutative ring with identity. Prove or disprove: the tensor product (over \(R\)) of two torsion free \(R\)-modules is torsion-free.

Determine, with proof, the maximal order of an element in a Sylow 3-subgroup of the symmetric group on 9 letters.

Let \(G\) be a finite group, let \(H\) be a subgroup of \(G\), and let \(\mathbb {C}[G]\) (respectively, \(\mathbb {C}[H]\)) denote the complex group algebra of \(G\) (respectively, \(H\)). Let \(I_H^G\) denote the functor from the category of finite-dimensional \(\mathbb {C}[H]\) modules to the category of finite-dimensional \(\mathbb {C}[G]\) modules which, on the level of objects, is defined by \[ I_H^G(M)=\mathbb {C}[G]\otimes _{\mathbb {C}[H]}M. \] Prove or disprove: \(I_H^G\) is an exact functor.

Let \(F\) be a field with \(\mathbb {Q}\subset F\subset \mathbb {C}\), where \(F/\mathbb {Q}\) is a finite Galois extension. Let \(\alpha \in F\) and let \(f(X)\in \mathbb {Q}[X]\) be its minimal monic polynomial. Assume that the absolute value \(|\alpha |\) is 1, and that \(\operatorname {Gal}(F/\mathbb {Q})\) is abelian.

1. Show that \(F\) is closed under complex conjugation.
2. Prove that \(|\beta |=1\) for every complex root \(\beta \) of \(f(X)\).
3. Writing \(f(X)=X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0\), show that \(|a_i|\le 2^n\) for all \(i\) with \(0\le i<n\).
4. Prove that \(F\) contains only finitely many algebraic integers having absolute value 1 and deduce that each of these is a root of unity.

Let \(N\) be a normal subgroup of a finite group \(G\). Assume that the center of \(N\) is trivial and that any automorphism of \(N\) is inner. Show that there exists a normal subgroup \(H\) of \(G\) such that \(G\) is isomorphic to the direct product of \(H\) and \(N\).

Let \(R\) be a commutative ring with 1. Suppose \(I\) and \(J\) are two ideals of \(R\) such that \(I+J=R\). Prove that \(R/(IJ)\cong R/I\oplus R/J\).

Let \(N\) be a positive integer. Consider the natural map from \(\operatorname {SL}(2,\mathbb {Z})\) to \(\operatorname {SL}(2,\mathbb {Z}/N)\) obtained by reducing each matrix entry modulo \(N\). Prove or disprove: this map is surjective.

Let \(\mathbb {Z}[x]\) be the ring of polynomials in \(x\) with integer coefficients. Let \(R\) denote the subring consisting of all polynomials having their coefficients of \(x\) and \(x^2\) equal to 0.

1. Show that \(\mathbb {Q}(x)\) is the field of fractions of \(R\).
2. Compute the integral closure of \(R\) in \(\mathbb {Q}(x)\).
3. Does there exist a polynomial \(g(x)\in R\) such that \(R\) is generated as a ring by 1 and \(g(x)\)?

Let \(R\) be a unique factorization domain with quotient field \(k\). Let \(f\) be an element of \(R[x]\). Show that if \(f\) can be factored as a product of lower-degree polynomials in \(k[x]\), then \(f\) can be factored as a product of lower-degree polynomials in \(R[x]\).