189-570A: Higher Algebra I
Due: Monday, October 30.
Group B. You only need to do problems 1-5.
(a) Consider the map p:C[x,y] --> C[t] defined by
p(f(x,y)) = f(t2, t3).
Show that its kernel is a principal ideal. Show that the
image of p is the set of polynomials g such that g'(0)=0.
(b) Consider the map q:C[x,y] --> C[t] defined by
q(f(x,y)) = f(t2-t, t3-t2).
Show that its kernel is a principal ideal. Show that the image of
q is the set of polynomials g such that g(0)=g(1).
(c) Give an intuitive explanation of these results in terms of the geometry
of the variety attached to the kernels of p and q.
2. Determine all the ideals of the ring R[[t]] of formal power series with
3. Let R be a ring, and let I be an ideal of the polynomial ring
R[x]. Suppose that the lowest degree of a non-zero polynomial in I
is n and
that I contains a monic polynomial of degree n. Show that I is principal.
4. Let R be a ring, let p be a prime ideal of R, and let
Rp denote the localisation of R at p.
(a) Show that R has a unique maximal ideal m,
which is generated by
the image of p in Rp (under the natural homomorphism
from R to Rp).
(b) Show that Rp/m is isomorphic to the fraction field of
the integral domain
Let f1,...,fr and g1,...,gs be
collections of polynomials in C[x1,...,xn], and
let U (resp. V) be the variety attached to the ideal generated by
f1,...,fr (resp. g1,...,gs).
Prove that if U and V do not meet,
then (f1,..,fr,g1,...,gs) is the
ideal R generated by 1.
The radical of an ideal I is the set of elements r in R such that some
power of r is in I.
(a) Prove that the radical of I is an ideal.
(b) Prove that the varieties in Cn
defined by two ideals I and J of C[x1,...,xn]
are equal if and only
if the radicals of I and J are equal.
7. Prove or disprove the following:
(a) The polynomial ring Q[x,y] is a Euclidean domain.
(b) The ring Z[x] is a principal ideal domain.
8. Let R=Z[i] be the ring of Guassian integers.
(a) Show that R is a Euclidean domain with size function given by
s(a+bi) = a2 + b2.
(b) Let p be a rational prime (i.e., a prime number, generating a prime ideal of Z.)
(i) R/pR is isomorphic to the field Fp2
with p2 elements, if p=3 mod 4.
(ii) R/pR is isomorphic to Fp x Fp,
if p=1 mod 4.
(iii) R/2R is isomorphic to F2[e]/(e2).
(c) Show that a prime number p
can be written as a sum of two integer squares if and only if p=2 or p
is congruent to 1 modulo 4.
9. (Extra Credit).
(a) A non-commutative ring is said to be left-euclidean if
it is equipped with a size function s from R to the positive reals
such that, for all a,b in R, there exists q and r satisfying
a = qb+r, with r=0 or s(r) < s(b).
A left ideal in a
non-comutative ring R is a subset of R which is an R-module under
left multiplication by R.
Show that if R is left-euclidean, then every left ideal of R is principal.
(b) Let R be the set of quaternions of the form a+bi+cj+dk, where
a,b,c,d are either all integers of halves of odd integers.
Show that R is a (non-commutative!!) subring of the ring of Hamilton's
(c) Show that this ring R is left-Euclidean (and hence, that each left ideal of R is principal, by (a).)
(d) Show that the left ideal of R generated by a prime number p is always
properly contained in a proper left ideal of R.
(e) Use (d) and (c) to prove Lagrange's celebrated four squares theorem: every
positive integer can be written as a sum of 4 perfect squares. (E.g.: