HOMEWORK 10
Question 1. Suppose that F is a eld extension of K . Prove that the closure
operator algK (S ) = K (S ) alg F satises the following properties:
(a) If S F , then S algK (S ).
(b) If S , T F and S T , then algK (S ) algK (T ).
(c) If S F , then

Algebra 2, Homework 1
Due: March 8, 2012.
Problem 1:
Let F E be an algebraic eld extension and R a ring such that
F R E . Prove that R is eld.
Problem 2:
Let E = Q( 2, 3). Then E has the Q-basis cfw_1, 2, 3, 6. Find
1
a, b, c, d Q such that (1 + 2 + 3)

Algebra 2, Homework 1 Solutions
Problem 1:
Let F E be an algebraic eld extension and R a ring such that F R E .
Prove that R is eld.
It is enough to show that r1 R whenever 0 = r R. Let f (x) = xn +
a1 xn1 + + an1 x + an F [x] be the minimal polynomial fo

Algebra 2, Homework 2
Due: April 19, 2012.
Problem 1:
Let k be an algebraically closed eld and consider the set W =
cfw_(x, y, z ) A3 | x2 = y 3 and y 2 = z 3 . Show that W is Zariski closed
and nd I (W ) k [x, y, z ].
Problem 2:
Set I = (y 2 + 2xy 2 + x2

Algebra 2, Homework 2 Solutions
Problem 1:
Let W = cfw_(x, y, z ) A3 | x2 = y 3 and y 2 = z 3 where k is an algebraically
closed eld. Show that W is Zariski closed and nd I (W ) k [x, y, z ].
Set J = x2 y 3 , y 2 z 3 k [x, y, z ]. Then W = Z (J ) is Zari

Practice problems - Math 552
May 3, 2011
#1. Let R be a ring and, A mod R, and B R mod. Let A be a
submodule of A and B be a submodule of B . Show that (A/A ) R (B/B )
is isomorphic to (A R B )/C where C is the subgroup of A R B generated
by all a b and a

Solutions (and corrections) to Practice problems Math 552
May 5, 2011
#1. Let R be a ring and, A mod R, and B R mod. Let A be a
submodule of A and B be a submodule of B . Show that (A/A ) R (B/B )
is isomorphic to (A R B )/C where C is the subgroup of A R

HOMEWORK 1
Question 1. Suppose that n 5 and that
H = cfw_ Sym(n) | [cfw_1, 2] = cfw_1, 2 .
Prove that H is a maximal subgroup of Sym(n).
In particular, Sym(5) has a maximal subgroup of index 10.
Question 2. Suppose that the eld F is an algebraic extension

HOMEWORK 2
Question 1. Read Hungerford V.5, the section on nite elds.
Question 2. Let E = Q( 2, 3, 5).
(a) Prove that E is a Galois extension of Q.
(b) Compute AutQ E .
(c) Find an element E such that E = Q().
Question 3. Let E be the Galois extension of

HOMEWORK 3
Question 1. Suppose that E is a nite Galois extension of K with Galois group
G = AutK E . Prove that if F is a subeld such that K F E and H = AutF E ,
then
AutK F NG (H )/H.
=
Question 2. Suppose that f (x) Q is an irreducible quartic which has

HOMEWORK 4
Question 1. Compute the Galois group over Q of the polynomial
x6 + 22x5 9x4 + 12X 3 37x2 29x 15
(Hint: Reduce mod 2,3,5.)
Question 2. Let f (x) = x4 + ax2 + b Q[x] be an irreducible polynomial with
roots , and splitting eld K .
(a) Prove that t

HOMEWORK 5
Denition. If A and B are subsets of the group G, then the commutator subgroup
of A and B is dened by
[A, B ] = [a, b] | a A, b B .
Question 1. Suppose that A and B are subgroups of the group G. Prove that
[A, B ]
A iff B
NG (A).
Denition. If G

HOMEWORK 6
Question 1. Let R be an integral domain and for each maximal ideal M of R,
identify the localization RM with the corresponding subring of the quotient eld K
of R. Prove that
R=
cfw_RM | M is a maximal ideal of R .
Question 2. An element r of a

HOMEWORK 7
Throughout let f (x) Z[x] be a monic irreducible polynomial and let 1 , , n
be the distinct roots of f (x) in its splitting eld E . Identify G = AutQ E with the
corresponding subgroup of Sym(n). Dene
(i j )
=
and
D = 2 .
i<j
Question 1. Prove t

HOMEWORK 8
Question 1. If R is a Noetherian ring and : R S is a surjective ring homomorphism, then S is also Noetherian.
Question 2. If R is a Noetherian ring and T is a multiplicative subset of R, then
the ring T 1 R is also Noetherian.
Question 3. Let R

HOMEWORK 9
Question 1. Let F2 be the free group on two generators. Prove that if P F2
is an arbitrary subset, then there exists a nite subset Q P such that for all
homomorphisms , : F2 F2 , if Q =
Q, then P =
P.
Some hints:
(1) Regard F2 as the subgroup