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Canonical static concept record

Rational point

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Summary

For a scheme or variety $X$ over a field $K$ , a $K$ -rational point is a $K$ -morphism $\operatorname{Spec}(K)\to X$ ; for affine varieties this is equivalent to a solution with coordinates in $K$ .

Record metadata

Canonical name: Rational point
Concept ID: rational_point
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebraic geometry
Domains: Algebraic geometry, Number theory
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

scheme or variety over $K$

Data

$K$ -valued point

Canonically induces

An arithmetic solution

Concept sources

Wikipedia — Rational point — Rational point · encyclopedic reference · source ID wp-rational-point
nLab — Rational point — Rational point · mathematical reference · source ID nlab-rational-point
Milne — Algebraic Number Theory — Algebraic Number Theory · course text · source ID milne-algebraic-number-theory
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text
Poonen — Rational Points on Varieties — Rational Points on Varieties · monograph · source ID poonen-rational-points-varieties

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

No direct relations are authored in this direction.

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Rational point Local point

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_rational_point_to_local_point_canonical_embedding
Relation type: Canonical embedding canonical-embedding
Direction: source → target
Endpoint roles: source: Embeds canonically into; target: Contains a canonical copy of
Authored annotation: embed globally into every completion

Authored explanation

A global $K$ -point induces a $K_v$ -point for every place $v$ .

How to interpret this relation type

Map one structure injectively into another by the standard structure-preserving inclusion or representation. This records a canonical copy, not necessarily literal set containment.

Relation sources

Milne — Algebraic Number Theory — Algebraic Number Theory · course text · source ID milne-algebraic-number-theory

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