Local arithmetic and the $p$-adic world — Atlas of Fundamental Concepts

StoryLocal arithmetic and the $p$ -adic world

Global arithmetic becomes tractable by examining one place at a time. Valuations define arithmetic notions of size and divisibility; their valuation rings and residue fields record integral and reduced information. Completing a number field at an Archimedean or non-Archimedean place produces real, complex, or $p$-adic local fields. Discrete valuation rings and complete local rings supply the local algebra used throughout number theory and algebraic geometry.

Step 1 of 19Integers $\mathbb Z$

Preparing the atlas…

Select any concept, construction junction, or annotated relation by pointer, touch, search, or keyboard.

Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Atlas of Fundamental Concepts

Select a concept

Move selected concept

Stories & Views

Compare concepts

Guide to the Atlas