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Theorems · Definition · linear algebra

fieldOfFiniteDimensional

(F : Type u_1) →
  (K : Type u_2) →
    [inst : Field F] → [h : CommRing K] → [IsDomain K] → [inst_2 : Algebra F K] → [FiniteDimensional F K] → Field K

An integral domain that is module-finite as an algebra over a field is a field.

Defined in
Mathlib.LinearAlgebra.FiniteDimensional.Basic
Cited by
1 results in Mathlib
Foundations
Depth 124 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
FieldCommRingIsDomainAlgebraFiniteDimensional
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Polynomial.lift_of_splits · cited by 0Polynomial.lift_of_splitsCommRing · cited by 17173CommRingAlgebra · cited by 11388AlgebraField · cited by 7404FieldIsDomain · cited by 2196IsDomainFiniteDimensional · cited by 1854FiniteDimensionalDivisionRing · cited by 1062DivisionRingdivisionRingOfFiniteDimensional · cited by 2divisionRingOfFiniteDimen…DivisionRing.mul_inv_cancel · cited by 1DivisionRing.mul_inv_canc…DivisionRing.nnqsmul · cited by 1DivisionRing.nnqsmulDivisionRing.qsmul · cited by 1DivisionRing.qsmulDivisionRing.qsmul_def · cited by 1DivisionRing.qsmul_defDivisionRing.ratCast_def · cited by 1DivisionRing.ratCast_defDivisionRing.div_eq_mul_inv · cited by 0DivisionRing.div_eq_mul_i…DivisionRing.inv_zero · cited by 0DivisionRing.inv_zeroDivisionRing.nnqsmul_def · cited by 0DivisionRing.nnqsmul_deffieldOfFiniteDimensionalCITED BYCITES

Cites19

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Cited by1

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