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FOURTH PROBLEM SHEET FOR ALGEBRAIC NUMBER THEORY M3P15 AMBRUS PÁL not accessed √ 1. Let K = Q( −5). Factorise the following ideals of OK into a product of prime ideals: √ √ √ √ (1 + 2 −5), (3 + 2 −5, 3 − 2 −5), (7 + −5). √ 2. In this question K = Q( −17). Let P / OK be a prime ideal such that 2 ∈ P . (a) Find two elements a, b ∈ OK such that P = (a, b). (b) Prove that the class of P in the class group Cl(K) has order 2. (c) Deduce that O√ K is not an Euclidean domain. (d) Factorise (1√+ −17) into prime ideals in OK . 3. Let K = Q( −23). Show that the class group of K has order at most 3. √ In the next few problems K = Q( d) is a quadratic field, where d is a square-free √ √ integer. Let z 7→ z be the map K → K given by the rule x + y d = x − y d for every x, y ∈ Q. Let Cl(K) be the class group of OK . For every I / OK its conjugate ideal I is defined as: I = {z|z ∈ I}. 4. Show that associating to an ideal I ⊆ OK its conjugate ideal I preserves congruence classes of ideals modulo the group of principal ideals, so that conjugation is a well-defined operation on Cl(K). Prove that an element of Cl(K) is fixed by conjugation if and only if it has order at most 2 (that is, is represented by an integral ideal I such that I 2 is principal). 5. Suppose from now on that d < 0. Prove that any element of order at most 2 in Cl(K) can be represented by an ideal I = P1 P2 · · · Pn , where Pi are distinct prime ideals over some of the prime numbers ramified in K. (A prime ideal P is over a prime number p if p ∈ P . A prime number p is ramified in K if there is exactly one prime ideal of OK over p.) 6. Let I ⊆ OK be an ideal from problem 5. Determine when I is principal. 7. Conclude that if |d| is not a prime, then Cl(K) has an element of exact order 2, in particular OK is not a principal ideal domain. 8. Prove that the class numbers of quadratic fields can be arbitrary large. Finally here are some claims I did not prove in the lectures. 9. Let K be a field and let G be a finite subgroup of K ∗ . Show that G is cyclic. Date: March 6, 2017. 1 2 AMBRUS PÁL 10. Let R ⊆ C be a subring. Prove that the following are equivalent: (a) the ring R is a Dedekind domain and a UFD. (b) the ring R is a PID. (Hint: you need to use both factorisations (in prime ideals over Dedekind domains, and into prime elements over UFD-s) in the proof of (a) implies (b).)