Math 621  Problem Set #3  Due Wed., Feb. 20, 2002

1. Let A be a complete discrete valuation ring with fraction field K, let L be a finite separable extension of K, and let B be the integral closure of A in L. Let 𝔭 be the maximal ideal of A.

a) Let P, P′ be primes of B over 𝔭. Show that the P -adic topology on L is the same as the P′-adic topology on L. [Hint: Recall that if K is a complete field and V is a finite dimensional K-vector space, then there is a unique way to make V a Hausdorff topological K-vector space, viz. the product topology.]

b) Show that ideal P ⊂ B consists precisely of the elements α ∈ L such that α^{n} → 0 under the P -adic topology as n → ∞.

c) Conclude that there is a unique prime ideal of B lying over 𝔭 ⊂ A.

d) What happens if the discrete valuation ring A is no longer required to be complete?

2. For any integer n > 1 let Φ_{n}(x) = ∏_{i} (x − ζ_{n}^{i}), where the product ranges over 1 ≤ i ≤ n such that (i, n) = 1.

a) Show that Φ_{n}(x) ∈ ℤ[x].

b) Let p be a prime number. Write y = x − 1 and f(y) = Φ_{p}(x). Show that f(y) is an Eisenstein polynomial in y, and hence that f and Φ_{p} are irreducible over ℤ and over ℚ. [Hint: Write Φ_{p}(x) = (x^{p} − 1)/(x − 1).]

c) More generally, let q = p^{m} be a prime power and let K = ℚ(ζ_{q}) with n = [K : ℚ]. Show n ≤ ϕ(q). Show that Φ_{q}(x) = Φ_{p}(x^{q/p}), and hence that Φ_{q}(1) = p and that (p) = (1 − ζ_{q})^{ϕ(q)} as ideals in O_{K}. Using n = efr, deduce that the prime (p) is totally ramified in O_{K} of ramification index n = ϕ(q) (and unramified elsewhere), that Φ_{q}(x) is irreducible over ℚ, and that Gal(K/ℚ) ≈ (ℤ/q)^{∗}.

d) Still more generally, let n > 1 be arbitrary. Show that Φ_{n}(x) is irreducible over ℚ, that [ℚ(ζ_{n}) : ℚ] = ϕ(n), and that Gal(ℚ(ζ_{n})/ℚ) ≈ (ℤ/n)^{∗}. [Hint: Factor n into a product of prime powers. Using ramification behavior show that (a, b) = 1 ⇒ ℚ(ζ_{a}) ∩ ℚ(ζ_{b}) = ℚ. Use Galois theory to show that if K, K′ ⊂ ℚ̄ are Galois extensions of ℚ with groups G, G′ such that K ∩ K′ = ℚ, then KK′ is Galois over ℚ and its Galois group is G × G′.]

e) What happens if ℚ is replaced by a finite field k?

3. For 0 ≠ a ∈ ℤ, define |a|_{15} = \frac{1}{15^{n}} if 15^{n} divides a but 15^{n+1} does not. Define |0|_{15} = 0. Let ℤ_{15} be the completion of ℤ with respect to the metric d(x, y) = |x − y|_{15}.

a) Show that each element of ℤ_{15} can be written uniquely as a series ∑_{i=0}^{∞} a_{i}15^{i}, with each 0 ≤ a_{i} < 15.

b) Using the Chinese Remainder Theorem, show that ℤ_{15} ≈ ℤ_{3} × ℤ_{5}.

c) Under your isomorphism in (b), which elements of ℤ_{15} correspond to (1, 0) and to (0, 1) respectively? Are they zero divisors? Is ℤ_{15} a discrete valuation ring?

4. Let A ⊂ K and B ⊂ L be as in problem 1, with L Galois over K. Let 𝔭 ⊂ A and P ⊂ B be the maximal ideals of A, B, and let e be the ramification index at P over 𝔭. Let D ⊂ B be the different. Show that D = P^{e−1} if (p, e) = 1 and that D ⊂ P^{e} if p|e. [Hint: If the extension is totally ramified, the minimal polynomial of a uniformizer of B at P is an Eisenstein polynomial of degree e; now differentiate to find D. More generally, let K′ be the maximal unramified subextension, and use that L is totally ramified over K′.]

5. Find the different and discriminant of ℤ[\sqrt{7}] over ℤ in two ways: using derivatives, and using complementary sets.