ALGEBRA PRELIM EXAM January 5, 2005

Let \(\mathcal {N}\) denote the set of two-by-two real matrices with determinant zero, let \(\operatorname {SL}(2,\mathbb {R})\) denote the set of two-by-two matrices of determinant one, and finally let \(\operatorname {GL}(2,\mathbb {R})\) denote the set of two-by-two matrices with nonzero determinant.

1. Show that both \(\operatorname {GL}(2,\mathbb {R})\) and \(\operatorname {SL}(2,\mathbb {R})\) act on \(\mathcal {N}\) by conjugation.
2. Describe the orbits of \(\operatorname {GL}(2,\mathbb {R})\) on \(\mathcal {N}\).
3. Describe the orbits of \(\operatorname {SL}(2,\mathbb {R})\) on \(\mathcal {N}\).

Let \(R\) be a ring with identity. Suppose \[ 0\longrightarrow K\longrightarrow P\longrightarrow M\longrightarrow 0 \] and \[ 0\longrightarrow K'\longrightarrow P'\longrightarrow M\longrightarrow 0 \] are two short exact sequences of \(R\)-modules. Suppose further that \(P\) and \(P'\) are projective. Prove that \(K'\oplus P\cong K\oplus P'\) as \(R\)-modules.

Let \(A\) be an \(n\)-by-\(n\) diagonal matrix with coefficients in a field \(F\). Suppose the characteristic polynomial of \(A\) has the form \[ (x-c_1)^{d_1}(x-c_2)^{d_2}\cdots (x-c_k)^{d_k} \] with \(c_1,\ldots ,c_k\) distinct. Let \(V\) denote the space of all \(n\)-by-\(n\) matrices over \(F\) which commute with \(A\). Compute the dimension of \(V\) as a vector space over \(F\).

Let \(R\) be a finite ring with identity. Let \(a\) be a nonzero element of \(R\) which is not invertible. Prove that \(a\) is a zero divisor.

Suppose \(R\) is a commutative Noetherian ring. Let \(R[x]\) denote the ring of polynomial with coefficients in \(R\). Prove that \(R[x]\) is Noetherian.

Determine the number of subgroups of order 16 in the symmetric group \(S_6\).

Let \(F\) be a field such that the multiplicative group of \(F\) is cyclic. Prove that \(F\) is a finite field.

Prove or disprove that \(f(x)=x^5-x+1\) is irreducible over the rationals.

Let \(G_F\) denote the Galois group of \(x^3-3x^2-3\) over the field \(F\). Determine \(G_F\) when:

1. \(F\) is the complex numbers;
2. \(F\) is the real numbers;
3. \(F\) is the rational numbers; and
4. \(F\) is the integers modulo 2.

Prove that there are no simple groups of order 1365.