DEPARTMENT OF MATHEMATICS & STATISTICS
Preliminary Ph.D. Examination in Algebra
September 2, 1993
Determine the number of 3 × 3 invertible matrices in a finite field having q elements.
Can a line segment with length equal to the positive real fifth root of 2 be constructed
(given unit length) in a finite number of steps using straightedge and compass?
Let A be a commutative ring (with multiplicative identity).
Let M be an A-module. Let R = End(M) be the set of endomorphisms of M
(i.e., the set of A-module homomorphisms M → M ). Define operations on R
that make R an A-algebra (i.e., a ring with compatible A-module structure).
Is the set of ring endomorphisms (as opposed to A-module endomorphisms) of
the ring A a ring?
When A = Z , the ring of integers, find the endomorphism ring of the Z-module
Show that any group of order 20 has a non-trivial proper normal subgroup.
Prove that a finitely-generated torsion-free module over a principal ideal domain is
Determine the isomorphism class of each of the Sylow subgroups of the alternating
group A5, the group of “even” permutations of a set of cardinality 5.
Let ζ be a primitive 7th root of unity in the field of complex numbers, let K = Q(ζ),
and H = Q(α), where α = cos(2π/7) and Q denotes the field of rational numbers.
Show that H = K ∩R, where R is the field of real numbers.
Prove that K and H are both normal extensions of Q.
Determine the Galois groups Gal(K : Q), Gal(H : Q), and Gal(K : H).
For p a prime the ring of p-adic integers Zp is defined to be the inverse limit of the
unique ring homomorphisms
. . . → Z/pnZ → . . . → Z/p2Z → Z/pZ .
Let π denote the canonical ring homomorphism Zp → Z/pZ.
Show that an element of Zp is invertible in Zp if and only if its image under π
Show that the kernel of π is a maximal ideal of Zp.
Show that any proper ideal in Zp is contained in the kernel of π. (Hence, ker(π)
is the only maximal ideal.)
Show that the kernel of π i