p-adic Numbers and Valuations
Handbook guide to p-adic numbers and valuations with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
p-adic Numbers
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Proposition
Let I be a nonzero ideal of the Dedekind domain R. Then there is a nonzero ideal I′ such that II′ is a principal ideal (a). Moreover, if J is an arbitrary nonzero ideal of R, I′ can be chosen to be relatively prime to J.
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Corollary
A Dedekind domain with only finitely many prime ideals is a PID.
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Corollary
Let I be a nonzero ideal of the Dedekind domain R, and let a be any nonzero element of I. Then I can be generated by two elements, one of which is a.
Proof / verification strategy: Translate the element statement into ideals or quotient rings, establish both inclusions or implications, and use the strongest ring hypothesis actually available.
Definitions and Comments
A p-adic integer can be described in several ways. One representation is via a series x = a0 + a1p + a2p2 + · · · , ai ∈Z. (1) (Let’s ignore the problem of convergence for now.) The partial sums are xn = a0 + a1p + · · · + anpn, so that xn −xn−1 = anpn. A p-adic integer can also be defined as a sequence of integers x = {x0, x1, . . . , } satisfying xn ≡xn−1 mod pn, n = 1, 2, . . . .
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Proposition
The p-adic integer x = {xn} is a unit of θp (also called a p-adic unit) if and only if x0 ̸≡0 mod p. In particular, a rational integer a is a p-adic unit if and only if a ̸≡0 mod p.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Corollary
Every nonzero p-adic integer has the form x = pnu where n ≥0 and u is a p-adic unit. Consequently, θp is an integral domain. Furthermore, θp has only one prime element p, and every x ∈θp is a power of p, up to multiplication by a unit.
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Definition
The quotient field Qp of θp is called the field of padic numbers. By (7.9.3), each α ∈Qp has the form pmu, where m is an integer (possibly negative) and u is a unit in θp. Thus α has a “Laurent expansion” a−r pr + · · · + a−1 p + a0 + a1p + · · · . Another representation is α = x/pr, where x is a p-adic integer and r ≥0.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Definition
The p-adic valuation on Qp is defined by vp(pmu) = m. In general, a valuation v on a field F is a real-valued function on F \ {0} satisfying: (a) v(xy) = v(x) + v(y); (b) v(x + y) ≥min(v(x), v(y)). By convention, we take v(0) = + ∞. The representation x = pmu shows that vp is indeed a valuation on Qp.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Proposition
Let F be the quotient field of an integral domain R. The absolute value | | on F is nonarchimedian if and only if |n| ≤1 for every integer n = 1 ± · · · ± 1 ∈R.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Proposition
If | | is a nonarchimedian absolute value, then |x| ̸= |y| implies |x + y| = max(|x|, |y|).
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Corollary With respect to the metric induced by a nonarchimedian absolute value,
With respect to the metric induced by a nonarchimedian absolute value, all triangles are isosceles.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Definitions and Comments
Let | | be a nonarchimedian absolute value on the field F. The valuation ring of | | is θ = {x ∈F : |x| ≤1}. In the p-adic case, θ = {x ∈Qp : vp(x) ≥0} = θp. By properties (ii) and (iv) of (7.9.5), θ is a subring of F.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Quick-reference relationships
Problem-solving workflow
Identify the ambient domain and extension
State the base ring or field and the integral elements under discussion.
Translate arithmetic into ideals or linear maps
Norms, traces, discriminants and ideal factorisation encode arithmetic structurally.
Track divisibility and integrality
Do not confuse being algebraic with being integral, or element factorisation with ideal factorisation.
Use local information where appropriate
Valuations and p-adic ideas measure divisibility using a topology different from the usual absolute value.
Check finiteness hypotheses
Finite generation, Noetherian conditions and finite extensions are frequently essential.
Verify with quadratic or integer examples
Use small extensions and ideals to test signs, degrees and multiplicities.
Worked-solution emphasis from the supplied source
The supplied worked solutions for this section repeatedly test ideal, root, field, prime, norm, Tor. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- Using the ordinary absolute value when a p-adic valuation is intended.
- Confusing element factorisation with ideal factorisation.
- Assuming integrality without a monic polynomial relation.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 7.9 | p-adic Numbers | 151–155 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.