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(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 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 f₁,...,f_r and g₁,...,g_s be collections of polynomials in C[x₁,...,x_n], and let U (resp. V) be the variety attached to the ideal generated by f₁,...,f_r (resp. g₁,...,g_s). Prove that if U and V do not meet, then (f₁,..,f_r,g₁,...,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 Cⁿ defined by two ideals I and J of C[x₁,...,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) = a² + b².

(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 p² 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 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 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=1²+1²+1²+ 0², 7=2²+1²+1² +1², etc...)