Harvard Math 129: Algebraic Number Theory Homework Assignment 4

William Stein

Due: Thursday, March 10, 2005

The problems have equal point value, and multi-part problems are of the same value. In any problem where you use a computer, include in your solution the exact commands you type and their output. You may use any software, including (but not limited to) MAGMA and PARI.

1. Let p be a prime. Let O_{K} be the ring of integers of a number field K, and suppose a ∈ O_{K} is such that [O_{K} : ℤ[a]] is finite and coprime to p. Let f(x) be the minimal polynomial of a. We proved in class that if the reduction \overline{f} ∈ 𝔽_{p}[x] of f factors as

\overline{f} = ∏ g_{i}^{e_{i}},

where the g_{i} are distinct irreducible polynomials in 𝔽_{p}[x], then the primes appearing in the factorization of pO_{K} are the ideals (p, g_{i}(a)). In class, we did not prove that the exponents of these primes in the factorization of pO_{K} are the e_{i}. Prove this.

2. Let a_{1} = 1 + i, a_{2} = 3 + 2i, and a_{3} = 3 + 4i as elements of ℤ[i].

(a) Prove that the ideals I_{1} = (a_{1}), I_{2} = (a_{2}), and I_{3} = (a_{3}) are coprime in pairs.

(b) Compute #ℤ[i]/(I_{1}I_{2}I_{3}).

(c) Find a single element in ℤ[i] that is congruent to n modulo I_{n}, for each n ≤ 3.

3. Find an example of a field K of degree at least 4 such that the ring O_{K} of integers of K is not of the form ℤ[a] for any a ∈ O_{K}.

4. Let 𝔭 be a prime ideal of O_{K}, and suppose that O_{K}/𝔭 is a finite field of characteristic p ∈ ℤ. Prove that there is an element α ∈ O_{K} such that 𝔭 = (p, α). This justifies why PARI can represent prime ideals of O_{K} as pairs (p, α). (More generally, if I is an ideal of O_{K}, we can choose one of the elements of I to be any nonzero element of I.)

5. (*) Give an example of an order O in the ring of integers of a number field and an ideal I such that I cannot be generated by 2 elements as an ideal. Does the Chinese Remainder Theorem hold in O? [The (*) means that this problem is more difficult than usual.]