PH.D. QUALIFYING EXAM SPRING 2012 - ALGEBRA
Answer all the questions. In each case justify your answer. Here Fpis the finite field with p
elements.
1. Let Gbe a group of order pn, where pis a prime.
(i) Show that the center of Gis non-trivial.
(ii) Show that every maximal subgroup of Gis normal.
2. Let Gbe a group of order pnm, where pis a prime and gcd(p, m) = 1. Show that G
has a subgroup of order pn.
3. Let Gbe the group with presentation
G=ha, b|a9, b4, bโ1ab =a5i.
(i) Find the order of G.
(ii) Find the center Zof G.
(iii) Find G/Z.
4. (i) Show that every a vector space has a basis. (Do not assume that Vis finite
dimensional.)
(ii) Let Vbe a vector space of finite dimension and let T:VโVbe a linear
transformation. Show that V=KโWwhere K= ker(T) and Wโผ
=Image(T).
5. Show that if a finite ring Rwith 1 admits an injective (ring) homomorphism from a
field, then the number of elements of Rmust be a power of a prime number. Is R
necessarily a field?
6. For a field Kand nโฅ1 let Jn,K denote the nรnmatrix over Kwhose (i, j) entry is
equal to (โ1)i+jโK.
(a) Find the Jordan form of J3,F2;
(b) Find the Jordan form of J3,F3.
(c) Find the Jordan form of J3,Q.
7. Let Rbe a PID. Show that every non-zero prime ideal is maximal.
8. Find the Galois group of the polynomial f(x) = x3โ3xโ1โQ[x].
9. Let Fbe a field and let f(x)โF[x]. Show that F[x]/(f(x)) is a field if and only if
f(x) is irreducible over F.
10. An R-module Mis said to be irreducible if M6={0}and Mhas no R-submodules
except {0}and M. Let Vbe a finite-dimensional vector space over a field k, and let
T:VโVbe a linear transformation. Then Tgives Vthe structure of a k[x]-module.
Prove that Vis irreducible as a k[x]-module if and only if the characteristic polynomial
of Tis irreducible in k[x].