Math 621  Problem Set #2  Due Mon., Feb. 11, 2002

1. Let A be a complete discrete valuation ring with respect to an absolute value | · |. Let K be its fraction field, with algebraic closure K̄. Let n be a positive integer.

a) Show that ∀ε > 0 ∃δ > 0 such that if a_{0}, …, a_{n−1} ∈ A satisfy |a_{i}| < δ for all i then the roots of x^{n} + a_{n−1}x^{n−1} + ⋯ + a_{1}x + a_{0} (in K̄) each have absolute value < ε.

b) More generally, let f(x) = x^{n} + a_{n−1}x^{n−1} + ⋯ + a_{0} ∈ A[x] be a polynomial with roots α_{j} ∈ K̄ and multiplicities r_{j}. Show that ∀ε > 0 ∃δ > 0 such that if b_{0}, …, b_{n−1} ∈ A satisfy |b_{i} − a_{i}| < δ for all i, then for every root β ∈ K̄ of g(x) = x^{n} + b_{n−1}x^{n−1} + ⋯ + b_{0} there is a j such that |β−α_{j}| < ε. [Hint: Show |f(β)| is small, and use f(x) = ∏(x−α_{j})^{r_{j}}.]

c) In the notation of (b), show that if ε is sufficiently small, then the number of β’s within ε of α_{j} (counting multiplicity) is equal to r_{j}. [Hint: Let g approach f, and factor each over K̄.]

d) Deduce that if f is separable and if ε is sufficiently small, then g is also separable.

2. a) Explicitly find the unramified extension of ℚ_{7} of degree 3. Also find a totally ramified Galois extension of ℚ_{7} of degree 3.

b) Do the same with ℚ_{7} replaced by 𝔽_{7}((t)).

3. Find the different and discriminant of ℤ[ζ_{7}] over ℤ. Which primes of ℤ divide the discriminant? (Hint: What are the factors of x^{7} − 1?)

4. a) Find the different and discriminant of ℂ[x, y]/(x^{2} + y^{2} − 1) over ℂ[x]. Describe the situation geometrically.

b) Do the same for k[x, y]/(y^{p} − y − x) over k[x], where k is a field of characteristic p.

5. Which of the following algebraic numbers α have the property that ℚ(α) lies in a field of the form ℚ(ζ_{n}): α = \sqrt{2}, \sqrt{5}, ^{3} \sqrt{2}. Justify your assertions directly. For each such α which lies in a ℚ(ζ_{n}), find n and express α in terms of ζ_{n}.

6. For each of the fields K = ℚ(\sqrt{−6}), ℚ(\sqrt{−5}), show that 6 has two distinct factorizations into irreducible elements in O_{K}, and use that to find a non-principal ideal I in O_{K}. Check that I^{2} is principal in O_{K}, and find a degree 2 extension L of K such that I becomes principal in O_{L}. Is O_{L} ramified over O_{K}?

7. Let R be an integral domain with fraction field K. Recall that a fractional ideal of R is an R-module M contained in K, with the property that cM ⊂ R for some non-zero c ∈ R.

a) Find all the fractional ideals of ℤ.

b) Show that the natural map

{equivalence classes of ideals} → {equivalence classes of fractional ideals} = C_{K}

is a bijection. (Here, we say I, J are equivalent if aI = J for some a ∈ K^{∗}.)

c) Deduce that R is a PID if and only if the class group C_{K} is trivial — i.e. if and only if every fractional ideal is principal. Under what circumstances is this also equivalent to R being a UFD?