Theorems · Inductive type · field theory

Field

Type u → Type u

A Field is a CommRing with multiplicative inverses for nonzero elements. An instance of Field K includes maps ratCast : ℚ → K and qsmul : ℚ → K → K. Those two fields are needed to implement the DivisionRing K → Algebra ℚ K instance since we need to control the specific definitions for some special cases of K (in particular K = ℚ itself). See also note [forgetful inheritance]. If the field has positive characteristic p, our division by zero convention forces ratCast (1 / p) = 1 / 0 = 0.

Defined in

Mathlib.Algebra.Field.Defs

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Foundations

Depth 0 from the axioms, rests on 1 definitions · uses no axioms