Math 254A, UC Berkeley, Fall 2001 (Kedlaya) Lecture 14: The p-adic Numbers September 28, 2001

Reference. Neukirch, Sections 2.1 and 2.2.

Outline of lecture.

1. Define the p-adic integers ℤ_{p} naively, as “infinite series in p.” Demonstrate addition, multiplication, additive inverses, multiplicative inverses. Define the p-adic numbers ℚ_{p} as infinite series in p with finitely many places after the decimal point.

2. Define the p-adic integers less naively, as coherent sequences in ℤ/pℤ × ℤ/p^{2}ℤ × · · ·.

3. Define the p-adic integers (resp., the p-adic numbers) even less naively, as the completion of ℤ (resp., of ℚ) under the p-adic valuation. (We’ll repeat this more generally in the next lecture.)

4. Notice that ℤ_{p} is a local ring with maximal ideal pℤ_{p}. (That is, every element of ℤ_{p} \ pℤ_{p} is a unit.) In particular, any rational r/s, where s is coprime to p, is a p-adic integer (in fact, a terminating or rational one, and conversely).

5. Demonstrate some simple cases of Hensel’s lemma (simplest form: any monic polynomial over ℤ_{p} with a simple root modulo p has a root in ℤ_{p}). For example, if n is a quadratic residue modulo p, then n has two square roots in ℤ_{p}. Consequence: the p-adic numbers include lots of algebraic numbers!

6. Note that since ℤ_{p} is complete with respect to the p-adic valuations, we can define some functions in terms of convergent power series. For example, if p ≠ 2, then exp(x) = ∑_{n=0}^{∞} x^{n}/n! and log(1 + x) = ∑_{n=1}^{∞}(−1)^{n−1}x^{n}/n converge for x ∈ pℤ_{p}

7. Mention that a polynomial F (x_{1}, …, x_{n}) has a zero in the p-adic integers if and only if it has a zero modulo every power of p.