**189-570A:** Higher Algebra I

## Assignment 4

## Due: Monday, October 30.

**Group B**. You only need to do problems 1-5.

**1**.
(a) Consider the map p:C[x,y] --> C[t] defined by

p(f(x,y)) = f(t^{2}, t^{3}).

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(t^{2}-t, t^{3}-t^{2}).

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
real coefficients.

**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
R_{p} 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 R_{p} (under the natural homomorphism
from R to R_{p}).

(b) Show that R_{p}/m is isomorphic to the fraction field of
the integral domain
R/p.

**5**.
Let f_{1},...,f_{r} and g_{1},...,g_{s} be
collections of polynomials in C[x_{1},...,x_{n}], and
let U (resp. V) be the variety attached to the ideal generated by
f_{1},...,f_{r} (resp. g_{1},...,g_{s}).
Prove that if U and V do not meet,
then (f_{1},..,f_{r},g_{1},...,g_{s}) is the
ideal R generated by 1.

**6**.
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 C^{n}
defined by two ideals I and J of C[x_{1},...,x_{n}]
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) = a^{2} + b^{2}.

(b) Let p be a rational prime (i.e., a prime number, generating a prime ideal of Z.)
Show that

(i) R/pR is isomorphic to the field F_{p2}
with p^{2} elements, if p=3 mod 4.

(ii) R/pR is isomorphic to F_{p} x F_{p},
if p=1 mod 4.

(iii) R/2R is isomorphic to F_{2}[e]/(e^{2}).

(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
quaternions.

(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.:
3=1^{2}+1^{2}+1^{2}+ 0^{2},
7=2^{2}+1^{2}+1^{2} +1^{2},
etc...)