MATH 732, SPRING 2009
HOMEWORK SET 8 sample solutions
A. Consider the triple (O, M, U ) whose objects O are groups, whose morphisms are group
homomorphisms f : A B with the property im f B , and whose forgetful functor U
is the usual one. Is this a (concr
MATH 732, SPRING 2009
HOMEWORK SET 7 sample solutions
A. Let R S be integral domains such that S is integral over R and let I be a nonzero
ideal of S . Show that I R = 0 .
Proof. Let i be a nonzero element of I . Since i is integral over R , it is a root
MATH 732, SPRING 2009
HOMEWORK SET 6 sample solutions
A. Suppose R, R are integral domains with quotient elds Q, Q respectively. Let : R R
be a ring homomorphism.
Find necessary and sucient conditions for there to exist a eld homomorphism : Q
r
(r)
Q wit
MATH 732, SPRING 2009
HOMEWORK SET 5 sample solutions
A. Let A be a set, a binary operation on A , and an equivalence relation on A that is
compatible with in the sense that a b , c d implies a c b d .
For a A , let a denote the equivalence class of a rel
MATH 732, SPRING 2009
HOMEWORK SET 4 sample solutions
A. Decide which of the following partially ordered sets are isomorphic. Give a good reason
for your answers, both positive and negative, but do not give a formal proof.
(1) (N, ) .
(2) (N3 , lex ) .
(3
MATH 732, SPRING 2009
HOMEWORK SET 3 sample solutions
A. Suppose cfw_X : is a collection (possibly innite) of subsets of F k and cfw_F :
is a collection (possibly innite) of subsets of F [x] .
(a) Show that I ( X ) = I (X ) .
Proof. Since each X is cont
MATH 732, SPRING 2009
HOMEWORK SET 2 sample solutions
A. Let F be a eld, let R, S be F -algebras, and let X be a basis of R . Suppose : R S
is a linear transformation. Prove that is an F -algebra homomorphism if and only if:
(i) (1R ) = 1S and (ii) (xy )
MATH 732, SPRING 2009
HOMEWORK SET 1 sample solutions
A. In this problem we consider the equation x3 3x + 1 = 0 over Q ; note f = x3 3x + 1
is irreducible in Q[x] . Let E be a splitting eld over Q for f .
Note. One can see f is irreducible either by notin
MATH 732, SPRING 2009
FINAL EXAM sample solutions
A. Let C be a denominator set in the commutative ring R , let S be another commutative
ring and let g, h : RC 1 S be ring homomorphisms such that g (r/1) = h(r/1) for
each r R . Prove that g = h .
Proof. L