Mathematics Branches, Topics, and Sub-Topics

A structured visual guide to the major mathematical areas and their relationships.

Search by code, branch, topic, subtopic, or a keyword from the descriptions.

12Kxx Generalizations of field theory

This subtopic studies generalizations of field theory, including broader algebraic systems where field-like methods still provide insight into structure and extension problems.

Specific topics

12K05 Near-fields

Overview

12K05 develops near-fields within generalizations of field theory. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.

Related Wikipedia Page

Near-fields (Wikipedia)

Useful Links

Key Ideas

  • Canonical examples and equivalent formulations for near-fields
  • Interactions between ring/field structure, morphisms, and module behavior
  • Common proof patterns and invariants used in modern algebraic practice

Typical Uses

Used to classify algebraic structures, transfer properties across extensions and localizations, and organize arguments in commutative algebra and algebraic geometry.

Applications

  • Core theoretical developments in algebra and arithmetic geometry
  • Symbolic computation and algorithmic algebra workflows
  • Foundational support for logic, coding theory, and number theory links

References

Recommended Textbooks

12K10 Semifields

Overview

12K10 develops semifields within generalizations of field theory. It emphasizes foundational definitions, structural theorems, and techniques that support both proof-oriented and computational workflows.

Related Wikipedia Page

Semifields (Wikipedia)

Useful Links

Key Ideas

  • Canonical examples and equivalent formulations for semifields
  • Interactions between ring/field structure, morphisms, and module behavior
  • Common proof patterns and invariants used in modern algebraic practice

Typical Uses

Used to classify algebraic structures, transfer properties across extensions and localizations, and organize arguments in commutative algebra and algebraic geometry.

Applications

  • Core theoretical developments in algebra and arithmetic geometry
  • Symbolic computation and algorithmic algebra workflows
  • Foundational support for logic, coding theory, and number theory links

References

Recommended Textbooks