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189-571B: Higher Algebra II

Assignment 1. Due: Wednesday, February 20.




1. Let p and q be two primes that are congruent to 3 modulo 4. Show that Z[p] and Z[q] are the rings of integers of Q(p) and Q(q) respectively, but that Z[p,q] is not the ring of integers of Q(p,q). Compute the ring of integers of this biquadratic field.

2. Let k be a field and let A=k[x2,x3] be the subring of the polynomial ring k[x] generated by x2 and x3. Show that A is Noetherian and of Krull dimension one, but that A is not integrally closed.

3. Let p be a prime and let K=Q(ζ) be the field generated over Q by a primitive p-th root of unity ζ.

(a) Show that the ring OK:=Z[ζ] is the ring of integers of K.

(b) For each rational prime q of Z, show that the ideal qOK of OK factors as qOK=(q1qt)e, where each qi is a prime ideal of residue degree f, and tef=p1.

(c) Give an explicit formula for t, f and e in terms of q (and p, of course).

4. Kunz, exercise 4, page 9.

5. Kunz, exercise 5, page 9.

6. Kunz, exercise 1, page 15.

7. Kunz, exercise 6, page 15.

8. Kunz, exercise 9, page 16.

9. Kunz, exercise 1, page 21.

10. Kunz, exercise 5, page 22.