Algebraic number theory M 3/4 P 15

Question Sheet 2

This is assessed work, due on the 25th of February.

1. Find all the prime numbers which are ramified in the following quadratic fields: (a) Q(\sqrt{10}), (b) Q(\sqrt{−7}), (c) Q(\sqrt{−1}), (d) Q(\sqrt{11}).

2. Find a positive integer a and the subset {b_{1}, …, b_{n}} ⊂ Z/a of residues modulo a such that a prime number p is split in Q(\sqrt{10}) if and only if p is congruent to one of the b_{i} modulo a. [Hint: reduce this to the computation of the Legendre symbol (\frac{10}{p}), then use its properties stated in the lectures.]

3. (a) We proved in the lectures that if a prime number p is split in Q(\sqrt{d}), then there are precisely two prime ideals in A(d) lying over it. Call them I_{1} and I_{2}. Show that their product I_{1}I_{2} is the principal ideal (p) ⊂ A(d). [Hint: use the fact that the only prime ideals of A(d) lying over p are (\sqrt{d} − a, p) and (\sqrt{d} + a, p), where a^{2} is congruent to d modulo p.]

(b) We proved in the lectures that if a prime number p is ramified in Q(\sqrt{d}), then there is precisely only prime ideal of A(d) lying over p. Call it I. Prove that I^{2} is the principal ideal (p) ⊂ A(d). [Hint: use the fact that if p is odd or p = 2 and d is even, then I = (\sqrt{d}, 2); when p = 2 and d is congruent to 3 modulo 4, then I = (\sqrt{d} − 1, 2).]

4. (a) Let d be a square free integer such that A(d) is a principal ideal domain. (For example, this is the case when A(d) is a Euclidean domain, e.g. for d = −1, −2, −3, −7, −11.) Deduce from Question 3 (a) that an odd prime number p such that (p, d) = 1, is a norm of some z ∈ A(d) if and only if the Legendre symbol (\frac{d}{p}) equals 1.

Deduce from (a) for d = −1, d = −2, d = −3, respectively, that

(b) a prime p is of the form a^{2} + b^{2} if and only if p = 2 or p is congruent to 1 modulo 4,

(c) a prime p is of the form a^{2} + 2b^{2} if and only if p = 2 or p is congruent to 1 or 3 modulo 8,

(d) a prime p is of the form a^{2} + ab + b^{2} if and only if p = 3 or p is congruent to 1 modulo 3.

(Here a and b are integers.)