MASTER’S EXAM, ALGEBRA, JANUARY 2012
1. Let Sbe a subset of the inner product space V. Show that the set of vectors
orthogonal to Sforms a subspace of V.
2. Show that no set of 3 vectors can span R4.
3. Let Abe a square matrix with real entries. Show that two (non-zero) eigenvec-
tors of Awhich correspond to different eigenvalues must be linearly independent.
4. Let Zdenote the additive group of integers, and let G=Z×Zbe the direct
product of Zwith itself. Let Hbe the subgroup of Gdefined by
H={(2m+n, 3n)
m, n ∈Z}.
Prove that G/H ≡Z/6Z.
5. Let Gbe a group, and suppose that Mand Nare both normal subgroups of
G. Suppose that M∩N={e}, where eis the identity of G. Prove that for any
m∈Mand n∈N,mn =nm.
6. Let Gbe the alternating group A4. Find a Sylow p-subgroup of Gfor each
prime pdividing the order of G.
7. Recall that R[x] denotes the ring of polynomials in xwith real coefficients. Let I
be the principal ideal generated by the polynomial x2−1. Let ψ:R[x]→R×R
be defined by ψ(p(x)) = (p(1), p(−1)) and prove that R[x]/I ∼
=R×R.
8. Recall that a polynomial is monic if the leading coefficient is 1. Find (with
proof) the number of monic irreducible polynomials in F3[x] of degree 6. Here
F3is the field with three elements.
9. Calculate the seventy-second cyclotomic polynomial Φ72(x).
10. Given that the discriminant of the cubic polynomial ax3+bx2+cx +dis
b2c2−4ac3−4b3d−27a2d2+18abcd, calculate the Galois group over the rationals
of the following polynomials:
(a) x3+ 2x2+ 3x+ 4
(b) x3−2x2−x+ 1