Tentamenskrivning i Algebraisk talteori 03-01-18

Lösningar

F.1 Baker, p.10-11.

F.2 Generalisation of the idea in the proof that there are infinitely many primes q ≡ 1 (mod 4). Namely, we assume that there are only finitely many primes ≡ 1 (mod p), say q_{1}, ..., q_{n}. We then consider the number

N = (2 · ∏_{i=1}^{n} q_{i})^{p} + 1.

N is odd, hence not divisible by 2 or by any q_{i}. But if q is any prime dividing N, then ζ = ^{p} \sqrt{N − 1} is an element of order 2p in (Z/qZ)^{×}, so 2p divides the order of this group, namely q − 1. In other words, q ≡ 1 (mod 2p), equivalently q ≡ 1 (mod p), since q is odd.

F.3 (i) Proposition 24 from my 2000 lecture notes. (ii) See p.44-45 of Koblitz’ book ‘An introduction to number theory and cryptography’. I gave this as a handout in class.

F.4 (i) I will omit the detailed computations. With the help of Theorem 37 and relations (107), (108) from my 2000 lecture notes, you may compute that there are precisely three reduced forms, namely f_{1} = x^{2} + 9y^{2}, f_{2} = 2x^{2} + 2xy + 5y^{2} and f_{3} = 3x^{2} + 3y^{2}. Note that the third of these is an imprimitive form.

(ii) Denote the given form by f_{0}. Prop. 48(i) implies that the primes p represented by some form of discriminant −36 are those for which either 4p | −36 or (\frac{−36}{p}) = 1. The former condition is satisfied only by p = 3.

Since 36 is a perfect square, the latter condition is equivalent to (\frac{−1}{p}) = 1, and hence is satisfied if and only if p ≡ 1 (mod 4). The imprimitve form f_{3} represents only multiples of 3 and hence no prime other than 3 itself. It is also easy to see that f_{1} (resp. f_{2}) represents no number congruent to 2 (resp. 1) modulo 3. We conclude that f_{1} (resp. f_{2}) represents all primes congruent to 1 (resp. 5) modulo 12.

The form f_{0} is primitive and, modulo 3, is congruent to 2x^{2} + xy + 2y^{2}, from which we also easily see that it cannot represent any integer congruent to 1 modulo 3. Hence, f_{0} must be equivalent to f_{2} and represent all primes congruent to 5 modulo 12.

F.5 Approximately Satser 103 and 104 in my 2000 lecture notes.

F.6 Instead of writing out the solution, let me refer you to Theorem 4.21 in the book of Stewart and Tall, which is where I got this exercise from !!