Characteristic (mathematics)

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The term characteristic in mathematics typically stands for:

• Something specific, unique, intrinsic or invariant to the situation
• Something that completely describes the given situation

Related terms are invariant.

In commutative algebra/field theory

Characteristic of a ring: The smallest positive integer  such that . If no such positive integer exists, then the characteristic is said to be zero. A field either has characteristic zero or has characteristic equal to a prime number.

In group theory

Characteristic subgroup: A subgroup that is invariant (or, gets mapped to itself) under any automorphism of the group.

In algebraic topology/differential geometry

Characteristic class: A natural transformation from the vector bundle functor to the cohomology functor on a manifold; in other words, it assigns to every vector bundle over a manifold, a cohomology class, such that a certain naturality diagram commutes.

In measure theory/analysis

Characteristic function of a set, also called its indicator function, is a function that takes value 1 on the set and 0 outside it.