Algebraic number theory M 3/4 P 15

Question Sheet 1

This is assessed work, due on the 11th of February.

1. (a) Find the ring of integers A(d) in the following quadratic fields: Q(\sqrt{10}), Q(\sqrt{−7}), Q(\sqrt{−12}).

(b) Which of the following quadratic integers are units: 4 + \sqrt{17}, \sqrt{−1}, 2 + \sqrt{5}, 2 + \sqrt{−5}?

(c) Which of the following quadratic integers are irreducible: n + \sqrt{−5}, where n = 0, 1, 2, 3, 4, 5, 6, 7?

2. In the lectures we have seen that A(−1) is a Euclidean domain with norm as the size function. Show that the same method also works for A(−2). (In fact, it also works for d = −3, −7, −11.)

3. Application to a Diophantine equation. We know that if A(d) is a Euclidean domain, then every element is a product of finitely many irreducible elements, and that such a factorization is unique up to order and multiplication of irreducible factors by units. By the previous example this is the case for A(−2). We also know that the only units in it are ±1. Pierre Fermat stated that the only integer solutions of y^{2} + 2 = x^{3} are (3, ±5). Prove his theorem in the following steps:

(a) Show that y must be odd.

(b) Rewrite the equation as (y + \sqrt{−2})(y − \sqrt{−2}) = x^{3}. If a + b\sqrt{−2} is a common divisor of y + \sqrt{−2} and y − \sqrt{−2}, it divides their sum and difference. Deduce that y + \sqrt{−2} and y − \sqrt{−2} are coprime (they have no non-unit common divisors).

(c) Using the unique factorization in A(−2) conclude that y + \sqrt{−2} and y − \sqrt{−2} are cubes in A(d), say, y + \sqrt{−2} = (c + d\sqrt{−2})^{3}. Prove that the only solutions of this equation are c + d\sqrt{−2} = ±1 + \sqrt{−2}. Deduce Fermat’s statement.

4. Consider the ideal I = (3 + \sqrt{−5}, 7) ⊂ A(−5).

(a) Prove that I is not principal (in particular, I ≠ A(−5)). [Hint: in the lectures we showed that (1 + \sqrt{−5}, 2) is not principal.]

(b) Prove that I^{2} is principal. [Hint: show that I^{2} = (2 + 3\sqrt{−5})].