MATH 123. HANDOUT ON SEPARABILITY
Let K L be a nite eld extension. We say that it is separable if for some/any
choice of algebraic closure K K, the K-algebra K L is isomorphic as a ring to
K
the direct sum of several copies of K. Our goal is to prove the
MATH 123, FINAL EXAM
1. Let K be a eld and let K K1 and K K2 be nite eld extensions, with
K1 /K separable.
a. (5pts) Show that the ring K1 K2 is isomorphic to a (nite) direct sum of
K
elds
K i.
i=1,.,n
b. (bonus 5pts) Assume that K2 /K is also separable.
MATH 123. MINI-PROJECTS IN COMMUTATIVE
ALGEBRA.
I. Bonus problem (5pts). Let A be a Dedekind domain. Let M be a nitely
generated torsion-free A-module. Consider the A-module M := HomA (M, A).
(a) Show that it is also nitely generated and torsion-free.
(b)
MATH 123. HANDOUT ON ASSOCIATED PRIMES
Let A be a commutative ring, and M an A-module. We say that p Spec(A)
is an associated prime of M if there exists a submodule M M such that p is a
minimal element among the primes in supp(M ).
Theorem 1. Assume that
MATH 123, PROBLEM SET 2, DUE: FEB. 8.
Problems marked by (*) are optional1
1.(a) Let R be a ring, N a right R-module, and M1 and M2 be left R-modules.
Based on the universal property show that
N (M1 M2 )
R
(N M1 ) (N M2 ).
R
R
(I.e., assume the existence
MATH 123, MIDTERM 2 SOLUTIONS. TUE., APRIL 5
1. Let A be a commutative ring and M an A-module.
(a) 2.5 pts. Let S A be a multiplicative set. Recall that Spec(AS ) identies
with the subset of Spec(A) consisting of primes p, such that p S = . Show that
the
MATH 123, PROBLEM SET 3. DUE: FEB. 15
Problems marked by (*) are optional1
1. Give an example of a situation when N1 N2 is an injection of abelian groups
(=Z-modules) and M is another abelian group, such that the map
N1 M N2 M
Z
Z
is no longer injective.
MATH 123, PROBLEM SET 12. DUE: TUE, APRIL 26.
1. Let H1 , H2 be any pair of groups, and let k be an arbitrary ground eld.
(a) Show that for any pair of representations i of Hi , the map
H
H
1 1 2 2 (1 2 )H1 H2
is an isomorphism.
(b) Let i , i = 1, 2 be an
MATH 123, PROBLEM SET 10. DUE: TUE, APRIL 12.
Throughout this PS, A will be a commutative Noetherian ring and M an A-module.
1. Describe all prime ideals in the ring k[x, y]/x y. Hint: use the surjection
k[x, y] k[x, y]/x y and analyze what primes in Spec
MATH 123, PROBLEM SET 11. DUE: TUE, APRIL 19.
1. Let A be a Dedelind domain, and let M be a nitely generated module. Show
that the short exact sequence
0 M tors M M torsf ree 0
can be split.
Hint: we know by the theorem from class that for any maximal ide
MATH 123, PROBLEM SET 1, DUE: FEB. 1.
Notes:
(a) You dont have to write solutions for those of the problems that you have done
in the framework of another course. Just let our CA know.
(b) If some part of an argument is identical to what we did in class o