Algebraic number theory M 3/4 P 15

Question Sheet 3

Questions 1,2 and 3 of this sheet form a part of assessed work, due on the 17th of March.

1. Is it true that for every finite set of primes there exists a quadratic field Q(\sqrt{d}) ramified in exactly these primes? (Prove or give a counter-example.)

2. Find the fractional ideals which are the inverses of the following integral ideals I ⊂ A(d):

(a) d = −5, I = (2 + \sqrt{−5}),

(b) d = −6, I = (2, \sqrt{−6}),

(c) d = −6, I = (5, \sqrt{−6} − 2).

3. Factor the following integral ideals of A(d) into prime ideals:

(a) d = −11, I = (5),

(b) d = −5, I = (6),

(c) d = −7, I = (1 + \sqrt{−7}).

4. (non-assessed) Let d < 0 be a squarefree integer, divisible by at least two primes. Show that the class number of Q(\sqrt{d}) is greater than 1.

[Hint: Let p be an odd prime dividing d. Prove that (p, \sqrt{d}) is not principal.]

5. (non-assessed) Let d > 1 be a squarefree integer, divisible by a prime which is congruent to -1 modulo 4. Show that every unit has norm 1 (and never −1 !).