Problem Set #10
Due Friday, April 5, 2013
a) Find the degree of = 2 + 3 over Q, and also nd its minimal polynomial.
b) Do the same for = 3 + 3 2.
c) Is Q() normal over Q? Is Q( )?
2. Let F = C(x). For a C, view C(x a) as a eld extension of F .
Problem Set #9
Due Friday, March 29, 2013
1. Which of the following rings R are discrete valuation rings? For those that are, nd
the fraction eld K = frac R, the residue eld k = R/m (where m is the maximal ideal),
and a uniformizer . For the othe
Problem Set #8
Due Friday, March 22, 2013
1. Let k be a eld, and let R = k [x, y ].
a) Find the primary decompositions of the ideals (y 2 x2 ) and (y 2 x2 )2 . In each case,
nd the associated primes; determine whether the given ideal is irreducib
Problem Set #7
Due Friday, March 15, 2013
1. a) Let R be a Noetherian ring, I the set of proper ideals of R, and I0 a subset of I .
Let P be a property that ideals in I0 may or may not have. Suppose that one can show
the following condition:
Problem Set #6
Due Friday, March 1, 2013
1. a) Show that if 0 M M M 0 is an exact sequence of R-modules, then M
is Noetherian if and only if M and M are. [Hint: For the reverse implication, consider
problem 2 of Problem Set 2 and use the Five Lem
Problem Set #5
Due Friday, Feb. 22, 2013
1. Let M be a nitely generated R-module. Prove that the following conditions on M are
i) M is locally free over R (i.e. Mm is free over Rm for all maximal ideals m R).
ii) For every maximal ide
Problem Set #4
Due Fri., Feb. 15, 2013
1. Let R be a commutative ring, and let M, N, S be R-modules. Assume that M is nitely
presented and that S is at. Consider the natural map
: S R Hom(M, N ) Hom(M, S R N )
taking s (for s S and Hom(M, N ) to
Problem Set #3
Due Fri., Feb. 8, 2013
1. Let P be a nitely generated projective R-module.
a) Show that there is a nitely generated free R-module F , and an R-module K , such
that 0 K F P 0 is exact.
b) Show that there exists a homomorphism j :
Problem Set #2
Due Fri., Feb. 1, 2013
1. Suppose that
is a commutative diagram of R-modules, with exact rows.
a) Show that if 1 is surjective and 2 , 4 are injective, then 3 is injective.
Problem Set #1
Due Fri., Jan. 25, 2013
1. Which of the following R-modules are nitely generated? Which are free? Which are
R-algebras? Among the R-algebras, which are nitely generated as R-algebras?
a) R = Z, M = Z/5 Z/7
b) R = Z, M = (5) = ideal
Problem Set #13
Due Mon., April 22, 2013
1. Suppose k K is a separable eld extension of degree n.
a) Show that K k [x]/(f (x) for some irreducible polynomial f (x) k [x] of degree
n, and that K k K K [y ]/(f (y ) as K -algebras. [Hint: Identify e
Problem Set #12
Due Wed., April 17, 2013
1. Let p be a prime number.
a) Use Eisensteins Irreducibility Criterion to show that the polynomial
f (x) = xp1 + xp2 + + x + 1
is irreducible over Q. [Hint: First set y = x 1.]
b) Give another proof of th
Problem Set #11
Due Friday, April 12, 2013
1. Suppose that L is an algebraic extension of a eld K that is not separable. Let p be
the characteristic of K . Let F0 = Fp (t) and let F1 = F0 (t1/p ). Show that there is an
embedding : F1 L such that
For each of the following, either give an example, or else prove that none exists.
1. An integral domain R and a non-zero R-module M such that M := Hom(M, R) is 0.
2. A ring R and an R-module M that is torsion-free but not