Math 514a Algebraic Number Theory Fall 1995

Homework assignment 5:

For K a number field, the following standard notations will be used: D_{K} is the discriminant of K, r_{1} and r_{2} are the number of real embeddings and pairs of complex conjugate embeddings, h = #Pic O_{K} is the class number, w = #O_{K tor}^{×} is the number of roots of unity, and R is the regulator (to be defined later).

26. For each quadratic field with |D_{K}| ≤ 31, compute:

a) The class number h and the structure of the class group

b) The fundamental unit ϵ if D_{K} > 0.

c) The real number

\frac{2^{r_{1}}(2π)^{r_{2}}hR}{w|D_{K}|^{1/2}}

to ten decimal places. Here R = 1 if D_{K} < 0 and R = log |ϵ| if D_{K} > 0.

27. Let D be the discriminant of a quadratic field. Define a function χ of positive integers as follows: if p is prime, then

χ(p) = (\frac{D}{p} ) = \{ \begin{matrix}0 & if p|D \\ −1 & if D is not a square modulo p \\ 1 & if D is a non-zero square modulo p. \end{matrix}

Then extend χ by mulitplicativity. For a few D, compute the real number

∑_{n=1}^{∞} \frac{χ(n)}{n}

as accurately as you care to. (This series converges, but not absolutely.) Make a conjecture relating these numbers to those in the previous exercise, and prove your conjecture in the case D = −4.

28. Let K = Q(\sqrt{−23}). Suppose the rational prime p splits, so pO_{K} = ℘_{1}℘_{2}. Prove that ℘_{1} is principal if and only if p = a^{2} + ab + 6b^{2} for integers a and b and ℘_{1} is not principal if and only if 2p = a^{2} + ab + 6b^{2} for integers a and b. How does this generalize to other fields?

29. Prove that for any number field K, there exist infinitely many primes ℘ of K of degree 1, i.e., such that O_{K}/℘ ≅ F_{p}. (See Marcus, exercise 30, p. 91.)

30. Let K be the cubic field generated by a root of X^{3} − X + 1 and let L be its Galois closure. Find the discriminants, rings of integers, and class groups of K and L. Show that Q(\sqrt{−23}) ⊆ L and no prime of Q(\sqrt{−23}) ramifies in L. Thus relative discriminants can be 1.

31. For K as in the last problem, show that 23O_{K} = ℘^{2}℘′. If p ≠ 23 is a rational prime and (\frac{−23}{p}) = −1

then pO_{K} is a product of two primes. On the other hand, if (\frac{−23}{p}) = 1 then pO_{K} is a product of three

primes if and only if p = a^{2} + 23b^{2} and p is inert if and only if p ≠ a^{2} + 23b^{2}. (This one may be a bit challenging.)

32. If K is a cubic field with r_{1} = r_{2} = 1, show that the fundamental unit ϵ satisfies the bound 4ϵ^{3} + 24 ≥ |D_{K}|. Find ϵ when K = Q( ^{3} \sqrt{2}).

33. Let K be the biquadratic field Q(\sqrt{5}, \sqrt{−1}). Find the decomposition and inertia groups for primes over 2 and 5. Find the Frobenius elements of all primes other than 2 and 5 and show that (p, K/Q) depends only on p modulo some integer; what’s the best possible integer? Show that no prime of Q(\sqrt{−5}) ramifies in K. Find the fundamental unit in K (i.e., a generator of units modulo torsion).