MATH 123, PROBLEM SET 5
In this PS all vector spaces will be nite-dimensional
1. Let V be a vector space over a eld K, and let T : V V be a linear operator.
Let
Im(T ) := Im(T n ) and ker(T ) := ker(T n ).
nN
nN
(a) Show that Im(T ) ker(T ) V is an isomor
MATH 123, PROBLEM SET 8.
Throughout this PS, A will be a commutative ring.
1. Recall the notation: Spec(A) is the set of prime ideals in A. For an ideal I A,
we dene V (I) Spec(A) as cfw_p Spec(A), | I p.
(a) Show directly that V (I n ) = V (I).
(b) Show
MATH 123, PROBLEM SET 9
Throughout this PS, A will be a commutative ring and M an A-module.
1. Let S A be a multiplicative set. Show that for a short exact sequence of
A-modules 0 M1 M2 M3 0, the sequence
0 (M1 )S (M2 )S (M3 )S 0
is also short exact. Dedu
MATH 123, PROBLEM SET 4.
1. (a) Let F be a nite extension of Fp , generated by an element x, such that
n
n
xp 1 = 1. Show that every element y of F satises y p = y. (Hint: look at what
the binomial formula does in characteristic p.) Deduce that raising to
MATH 123: SAMPLE FINAL PROBLEMS
Problem 1 (Fall 2007, Day 1)
Let f (x) = x4 7 Q[x].
(1) Show that f is irreducible in Q[x].
(2) Let K be the splitting eld of f over Q. Find the Galois group of K/Q.
(3) How many subelds L K have degree 4 over Q? How many o
Problem Suggestion
Consider K = Q( 1) a nite eld extension of Q.
1. Compute the discriminant . What is its prime factorization?
2. Compute G = Gal(Q( 1)/Q).
3. Let OK = cfw_a + b 1|a, b Z. This is called the ring of algebraic integers of K. In our
case, i
MATH 123, PSET 7 (OPTIONAL)
1. (1pt) Let K L be a Galois extension, and let denote the Galois group
Gal(L/K). Fix an embedding : L K. Consider the ring K L, equipped with
K
the following pieces of structure:
K-algebra,
L-algebra,
The action of (via its
MATH 123, PROBLEM SET 8
Throughout this PS, A will be a commutative ring.
1. Recall the notation: Spec(A) is the set of prime ideals in A. For an ideal I A,
we dene V (I) Spec(A) as cfw_p Spec(A), | I p.
(a) Show directly that V (I n ) = V (I).
(b) Show t
MATH 123, PROBLEM SET 6
1. Let K L be a nite eld extension, and let K Ls L be the maximal separable subextension (see PS 4, Problem 6). Show that deg(Ls /K) equals
| Spec(K L)|.
K
2. Let A be a semi-simple algebra over an algebraically closed eld. Evident
MATH 123, PROBLEM SET 9
Throughout this PS, A will be a commutative ring and M an A-module.
1. Let S A be a multiplicative set. Show that for a short exact sequence of
A-modules 0 M1 M2 M3 0, the sequence
0 (M1 )S (M2 )S (M3 )S 0
is also short exact. Dedu
Practice Final
1. Let K = Q[x]/(x17 1).
(a) How many elds L are there with Q
L
K?
(b) How many of them are Galois over Q?
(c) Find a subeld L K of degree 2 over Q.
2. Recall that a eld extension F K is said to be algebraic if every element K satises a
pol