Math 254A, UC Berkeley, Fall 2001 (Kedlaya) Problem Set 2: Number Fields and Ideals Due in class Monday, September 10

Reference. Neukirch, Sections 1.2 and 1.3.

Jargon watch. An ideal of a ring of integers is sometimes called an integral ideal to emphasize that it is not just a fractional ideal.

Conjugates, Trace and Norm.

Please submit all of the following problems.

1. (A theorem of Kronecker) Let α be an algebraic integer all of whose conjugates in ℂ lie in the closed unit disc. Prove that α is a root of unity. (Hint: consider the powers of α.)

2. Let x be an element of a number field K such that Trace(x^{n}) is a rational integer for all n ∈ ℕ. Prove that x is an algebraic integer.

Ideals and Unique Factorization.

Please submit four of the following problems.

1. Find an example to show that a proper subring of the ring of integers of a number field need not have unique factorization of ideals. (Hint: this can already be done in a quadratic number field.)

2. Neukirch exercise I.3.3: Let d be squarefree and p a prime number not dividing 2d. Let 𝔬 be the ring of integers of ℚ(\sqrt{d}). Show that (p) = p𝔬 is a prime ideal of 𝔬 if and only if the congruence x^{2} ≡ d (mod p) has no solution in rational integers.

3. Show that ℤ[\sqrt{−5}] has exactly two ideal classes, the class of principal ideals and the class containing (2, 1 + \sqrt{−5}). (Hint: suppose 𝔞 is an ideal which is not principal and α is a nonzero element of 𝔞 of minimum norm. Prove that 2𝔞 ⊆ (α).)

4. Neukirch I.3.6: Prove that every ideal of a Dedekind domain can be generated by two elements. (Hint: use Neukirch I.3.5: The quotient of a Dedekind domain by a nonzero ideal is a principal ideal ring.)

5. Neukirch I.3.8: Let 𝔪 be a nonzero integral ideal of the Dedekind domain 𝔬. Show that in every ideal class of 𝔬, there exists an integral ideal prime to 𝔪.

6. Let K be any field and let L be a finite separable extension of the field K(x) of rational functions over K. Prove that the integral closure of K[x] in L is a Dedekind domain. (Hint: imitate the proof of Theorem I.3.1 in Neukirch.)