Algebra Qualifying Exam
January 2007
1. Let G be a group of order 21. Show that there is an element x ∈ G of order 7 such that
hxi ⊳ G.
7 pts.
2. Let G be a finite nilpotent group. Let M be a maximal proper subgroup of G. Show
that the index of M in G is prime.
7 pts.
3. (a) Define the cyclotomic polynomial φp (x) ∈ Z[x] where p is a prime number and
prove that it is irreducible over Q.
(b) Is x5 + x4 + x3 + x2 + x + 1 irreducible over Q?
10 pts.
4. (a) Let F ⊆ E be a field extension and α ∈ E an algebraic element over F . Define
irr(α, F ) (in any of the various equivalent ways).
p
√
10 pts.
(b) Let α = 1 − 3 ∈ C. Find irr(α, Q).
5. Let F be an arbitrary field.
(a) State and prove an equivalent condition for a polynomial f (x) ∈ F [x] to be such
that f ′ (x) = 0.
(b) Let p(x) ∈ F [x] be an irreducible polynomial. Prove that
p(x) has multiple roots ⇔ p′ (x) = 0 .
(c) Can you give an example of an irreducible polynomial p(x) ∈ Q[x] with multiple
roots?
14 pts.
6. Let G be a finite group. For each n ∈ Z+ , let
Tn (G) := {x ∈ G : xn = 1}.
Suppose that for every divisor n of |G| we have that |Tn (G)| ≤ n.
(a) Let |G| = pa11 · · · par r where the pi are distinct primes. Show that Tpai i (G) is the
unique pi -Sylow of G.
(b) Show that Tpai i (G) is cyclic.
(c) Deduce that G is cyclic.
(d) Let F be a finite field. Show that F \ {0} is a cyclic group under multiplication.
20 pts.
7. Let R be a commutative ring with 0 6= 1. Let P be a prime ideal of R and S := R \ P .
(a) Show that S is a submonoid of (R, ·).
(b) Consider the ring of fractions S −1 R. Let
x
I := { ∈ S −1 R : x ∈ P, s ∈ S}.
s
Show that I is the unique maximal ideal of S −1 R.
12 pts.
8. (a) Define the notion of projective module.
(b) Let d be a divisor of n. Describe a nontrivial Zn -module structure on Zd . Explain
why it is well-defined.
(c) Suppose gcd(d, n/d) = 1. Show that the Zn -module Zd is projective.
(d) Show that the Z4 -module Z2 is not projective.
20 pts.
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