Harvard Math 129: Algebraic Number Theory Homework Assignment 7

William Stein

Due: THURSDAY, April 14, 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 K be a number field. Prove that p | d_{K} if and only if p ramifies in K. (Note: This fact is proved in many books—finding a proof in one and rephrasing it in your own words and the context of this course is not cheating.)

2. Using Zorn’s lemma, one can show that there are homomorphisms Gal(\overline{ℚ}/ℚ) → {±1} with finite image that are not continuous, since they do not factor through the Galois group of any finite Galois extension. [Hint: The extension ℚ(\sqrt{d}, d ∈ ℚ^{∗}/(ℚ^{∗})^{2}) is an extension of ℚ with Galois group X ≈ ∏ 𝔽_{2}. The index-two open subgroups of X correspond to the quadratic extensions of ℚ. However, Zorn’s lemma implies that X contains many index-two subgroups that do not correspond to quadratic extensions of ℚ.]

3. (a) Give an example of a finite nontrivial Galois extension K of ℚ and a prime ideal 𝔭 such that D_{𝔭} = Gal(K/ℚ).

(b) Give an example of a finite nontrivial Galois extension K of ℚ and a prime ideal 𝔭 such that D_{𝔭} has order 1.

(c) Give an example of a finite Galois extension K of ℚ and a prime ideal 𝔭 such that D_{𝔭} is not a normal subgroup of Gal(K/ℚ).

(d) Give an example of a finite Galois extension K of ℚ and a prime ideal 𝔭 such that I_{𝔭} is not a normal subgroup of Gal(K/ℚ).

4. Let S_{3} by the symmetric group on three symbols, which has order 6.

(a) Observe that S_{3} ≅ D_{3}, where D_{3} is the dihedral group of order 6, which is the group of symmetries of an equilateral triangle.

(b) Use (4a) to write down an explicit embedding S_{3} ↪ GL_{2}(ℂ).

(c) Let K be the number field ℚ(\sqrt[3]{2}, ω), where ω^{3} = 1 is a nontrivial cube root of unity. Show that K is a Galois extension with Galois group isomorphic to S_{3}.

(d) We thus obtain a 2-dimensional irreducible complex Galois representation

ρ : Gal(\overline{ℚ}/ℚ) → Gal(K/ℚ) ≅ S_{3} ⊂ GL_{2}(ℂ).

Compute a representative matrix of Frob_{p} and the characteristic polynomial of Frob_{p} for p = 5, 7, 11, 13.