Theorems · Theorem · commutative algebra
isAlgebraic_iff_isIntegral
∀ {K : Type u} {A : Type v} [inst : Field K] [inst_1 : Ring A] [inst_2 : Algebra K A] {x : A},
IsAlgebraic K x ↔ IsIntegral K xAn element of an algebra over a field is algebraic if and only if it is integral.
- Defined in
- Mathlib.RingTheory.Algebraic.Integral
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 112 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Algebrastatement and proof · cited by 11,388
- Ringstatement and proof · cited by 7,463
- Fieldstatement and proof · cited by 7,404
- Polynomialproof · cited by 5,681
- MulZeroClass.zero_mulproof · cited by 1,625
- Polynomial.Cproof · cited by 1,598
- map_mulproof · cited by 1,137
- Polynomial.aevalproof · cited by 615
- Polynomial.leadingCoeffproof · cited by 498
- IsIntegralstatement · cited by 427
- IsAlgebraicstatement and proof · cited by 163
Cited by12
Results whose statement or proof uses this declaration.
- IsAlgebraic.isIntegralproof · cited by 25
- isAlgebraic_of_isFractionRingproof · cited by 6
- mem_algebraicClosure_iffproof · cited by 5
- Algebra.isAlgebraic_iff_isIntegralproof · cited by 3
- exists_isTranscendenceBasis_and_isSeparable_of_linearIndepOn_powproof · cited by 2
- IntermediateField.exists_algHom_adjoin_of_splits_of_aevalproof · cited by 1
- ax_grothendieck_of_locally_finiteproof · cited by 1
- Subalgebra.algebraicClosure_eq_integralClosureproof · cited by 0
- FunctionField.finiteDimensional_of_adjoin_transcendentalproof · cited by 0
- RatFunc.isAlgebraic_adjoin_simple_X'proof · cited by 0
- transcendental_aeval_iffproof · cited by 0
- IsFractionRing.isAlgebraic_iff'proof · cited by 0