PH. D. QUALIFYING EXAM FALL 2010 - ALGEBRA
Answer all the questions. For each question give appropriate proofs.
1. Let p, q be prime numbers. Show that a group of order p2is abelian and that a group
of order pq, p < q, cannot be simple.
2. (a) Let N, M be normal subgroups of the group G, where G=M N. Prove that
G/(M∩N)∼
=G/N ×G/M.
(b) Let m, n ∈N. Show that Zm×Zn∼
=Zmn if and only if gcd(m, n) = 1.
3. (a) List the non-isomorphic abelian groups of order 600?
(b) How many elements of order 2 does each of these groups have?
4. Let D8=ha, b|a2, b2,(ab)4ibe the dihedral group of order 8. Let H=hbi. Then
D8acts on the right cosets of Hin D8to give a permutation representation of D8as a
subgroup of S4. Let V= SpanC({e1, e2, e3, e4}) be the corresponding CD8permutation
module, where D8permutes the indices of the basis {e1, e2, e3, e4}. Show that there are
irreducible submodules W1, W2, W3of Vhaving dimensions 1,1,2 (respectively) such that
V=W1⊕W2⊕W3.
5. Let Rbe a principal ideal domain. Show that a non-zero element of Ris a prime if
and only if it is irreducible.
6. Let φ:V→Wbe a linear transformation of vector spaces (not necessarily finite-
dimensional). Show that the kernel (kerφ) and image (Im(φ)) of φare subspaces of Vand
W(respectively), and that
dim V= dim ker φ+ dim Im(φ).
7. Let F⊆K⊆Lbe fields. Prove: If the extensions L/K and K/F are algebraic, then
L/F is also algebraic.
8. Let Rbe a ring with 1 6= 0. Recall that an idempotent is an e∈Rsuch that e2=e.
Let ebe a central idempotent. Show that Re and R(1 −e) are 2-sided ideals of Rand that
R∼
=Re ×R(1 −e).
9. Fix r∈N. Determine all fields Q⊂K⊂Csuch that Gal(K/Q)∼
=Cr
2,where C2is
the cyclic group of order 2.
10. Determine, up to isomorphism, the Galois group of each of the following polynomials:
(1) x4−49 ∈Q[x] over Q;
(2) x4+ 49 ∈Q[x] over Q;
(3) x5−4x+ 2 ∈Q[x] over Q.
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