PH. D. QUALIFYING EXAM WINTER 2010 - ALGEBRA
Answer all the questions. Here Fp=Z/pZis the finite field with pelements and Fpnis
the finite field with pnelements.
1. Show that any finitely generated subgroup of the additive group Qis cyclic.
2. Prove Cauchy’s theorem: let Gbe a finite group where |G|=nand let pbe a prime
dividing n. Then there is an element in Gof order p.
3. Show that a group of order pqr, where p<q<rare primes, cannot be simple.
4. Let M∈GL(n, C) be a matrix of finite order. Show that Mis diagonalizable. Is the
same result true for matrices in GL(n, F ), where Fis any field?
5. Construct the field F=F125 with 125 elements. How many primitive elements are
there in the extension F/F5? Here, a primitive element is α∈Fsuch that F5(α) = F.
6. Let V=C[x], considered as a vector space over C. Let D=d2
dx2−d
dx be a linear
transformation of V. Let Wbe the smallest subspace of Vthat contains the element x3
and which is invariant under D. Find Wand then calculate the Jordan Canonical form of
D|Wand of D|W+I, where Iis the identity function on Wand D|Wis the restriction of
Dto W.
7. Let Abe a finite abelian group of order n=pkm, where pis a prime and gcd(p, m) = 1.
Show that
(Z/pkZ)⊗ZA
is isomorphic to the Sylow p-subgroup of A.
8. Prove that Z[x] is not a principal ideal domain.
9. Prove that in a unique factorization domain a non-zero element is a prime if and only
if it is irreducible.
10. Find the Galois group of:
(1) x4+ 1 ∈Q[x];
(2) x4+ 1 ∈R[x];
(3) the extension F16/F2.
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