Math 254A, UC Berkeley, Fall 2001 (Kedlaya) Problem Set 6: The p-adic Numbers/Completions Due in class Monday, October 8

Please submit eight of the following problems. If you took the 1997 Putnam exam, please do not submit More on Valuations 1.

References. Neukirch, Sections 2.1 and 2.2 for valuations and the p-adic numbers; Section 2.4 for completions.

More on Valuations.

1. (1997 Putnam competition) For each positive integer n, write the sum ∑_{m=1}^{n} \frac{1}{m} in the form \frac{p_{n}}{q_{n}}, where p_{n} and q_{n} are relatively prime positive integers. Determine all n such that 5 does not divide q_{n}.

2. (Manjul Bhargava) Let S be an infinite subset of ℤ and p a prime number. Pick a_{0} ∈ S, then recursively define a_{n} as an integer m ∈ S minimizing v_{p}(m−a_{0})+⋯+v_{p}(m−a_{n−1}). Prove that a polynomial with rational coefficients takes integer values on S if and only if it is an integer linear combination of the polynomials P_{n}(x) = ∏_{i=0}^{n−1}(x − a_{i})/(a_{n} − a_{i}). Optional bizarre consequence: the quantities v_{p}(a_{n} − a_{0}) + ⋯ + v_{p}(a_{n} − a_{n−1}) for n = 0, 1, … do not depend on any choices made in defining the a_{i}.

3. Let K be a number field. Prove that α ∈ K is an algebraic integer if and only if |α|_{𝔭} ≤ 1 for all nonzero prime ideals 𝔭 of 𝔬_{K}.

The p-adic Numbers.

1. Neukirch exercise II.1.1: A p-adic number a is a rational number if and only if its sequence of digits becomes periodic at some point. (Hint: write p^{m}a = b + cp^{l}/(1 − p^{n}) for 0 ≤ b < p^{l}, 0 ≤ c < p^{n}.)

2. Neukirch exercise II.2.4, modified: let ϵ ∈ 1 + pℤ_{p}. Show that the function from ℤ to ℤ_{p} defined by n ↦ ϵ^{n} extends to a function from ℤ_{p} to itself which is continuous with respect to the p-adic valuation. (Hint: one way to do this involves the binomial series (1 + x)^{n} = ∑_{i} \binom{n}{i}x^{i}.)

3. Neukirch exercise II.2.6: The fields ℚ_{p} and ℚ_{q} are not isomorphic unless p = q. (Hint: when do elements of ℚ_{p} have p-th roots?)

4. Neukirch exercise II.2.9 (p-adic Weierstrass Preparation Theorem): every nonzero power series f(x) = ∑_{n=0}^{∞} a_{n}x^{n} ∈ ℤ_{p}⟦x⟧ admits a unique representation f(x) = p^{m}P (x)U(x), where U(x) is a unit in ℤ_{p}⟦x⟧ and P (x) ∈ ℤ_{p}[x] is a monic polynomial satisfying P (x) ≡ x^{l} mod p, where l is the degree of P.

5. Here’s why not to define the n-adic numbers for n not prime. Let n be a composite positive integer, and define the ring ℤ_{n} as the set of sequences {x_{i}}_{i=1}^{∞}, where x_{i} ∈ ℤ/n^{i}ℤ and x_{i+1} ≡ x_{i} (mod n^{i}). Prove that:

(a) if n is a power of a prime p, then ℤ_{n} ≅ ℤ_{p};

(b) if n is not a power of a prime, then ℤ_{n} has zerodivisors.

Optional: prove that ℤ_{n} ≅ ∏_{p} ℤ_{p}, where the product runs over all primes dividing n.

Completions.

Throughout this section, let R be a complete discrete valuation ring with maximal ideal 𝔪. If you wish, you may take R to be the valuation ring of a finite extension of ℚ_{p}. You may even take R = ℤ_{p} if you wish.

1. (Newton’s iteration) Let P (x) be a monic polynomial over R and r_{0} ∈ R an element such that |P (r_{0})| < |P′(r_{0})|^{2}. Prove that the sequence {r_{n}}_{n=0}^{∞} given by the iteration r_{n+1} = r_{n} − P (r_{n})/P′(r_{n}) converges to a root r of P (x), and that v_{p}(r_{n} − p) ≥ 2^{n−c} for some c. This gives a much faster method for computing (approximate) roots of polynomials than does the proof of Hensel’s lemma.

2. Find a version of Newton’s iteration that allows you to compute the multiplicative inverse of a unit r ∈ R using only multiplications and additions, with a similar rate of convergence (i.e., you should need about log n steps to compute the inverse modulo 𝔪^{n}). Optional: implement your method in Magma.

3. (Eisenstein’s irreducibility criterion) Let P (x) = x^{n} + a_{1}x^{n−1} + ⋯ + a_{n−1}x + a_{n} be a polynomial over R such that a_{1}, …, a_{n} ∈ 𝔪 and a_{n} ∉ 𝔪^{2}. Prove that P (x) is irreducible over R. Optional: use this to prove the polynomial x^{p−1} + ⋯ + x + 1 is irreducible over ℤ.

4. Assume 𝔪 is generated by the prime number p ∈ ℤ and that the map x ↦ x^{p} on R/𝔪 is a bijection. (In other words, R/𝔪 is perfect. For example, this is true if R is the valuation ring of a finite unramified extension of ℚ_{p}.) Prove that for each x ∈ R/𝔪, there exists a unique y ∈ R which reduces to x modulo 𝔪, such that y has a p^{n}-th root for n = 1, 2, . . .. The element y is called the Teichmüller lift of x. (Hint: prove that x^{p^{n}} modulo p^{n+1} depends only on x modulo p.)

5. Show that for every automorphism τ of R/𝔪, there exists a unique automorphism of R inducing τ on R/𝔪. (Hint: every element of R can be written as a power series ∑_{n=0}^{∞} p^{n}y_{n}, where each y_{n} is a Teichmüller lift.) If τ is the map x ↦ x^{p}, the resulting automorphism of R is called the Frobenius automorphism.