Download FOURTH PROBLEM SHEET FOR ALGEBRAIC NUMBER THEORY

FOURTH PROBLEM SHEET FOR ALGEBRAIC NUMBER
THEORY M3P15
AMBRUS PÁL
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1. Let K = Q( −5). Factorise the following ideals of OK into a product of prime
ideals:
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√
√
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(1 + 2 −5), (3 + 2 −5, 3 − 2 −5), (7 + −5).
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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.
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In the next few problems K = Q( d) is a quadratic field, where d is a square-free
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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.
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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).)