Theorems · Theorem · commutative algebra
IsIntegral.isAlgebraic
∀ {R : Type u} {A : Type v} [inst : CommRing R] [inst_1 : Ring A] [inst_2 : Algebra R A] [Nontrivial R] {x : A},
IsIntegral R x → IsAlgebraic R xAn integral element of an algebra is algebraic.
- Defined in
- Mathlib.RingTheory.Algebraic.Integral
- Cited by
- 18 results in Mathlib
- Foundations
- Depth 111 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Ringstatement and proof · cited by 7,463
- Polynomialproof · cited by 5,681
- Algebra.algebraMapproof · cited by 4,706
- Nontrivialstatement and proof · cited by 2,416
- Polynomial.Monicproof · cited by 461
- IsIntegralstatement and proof · cited by 427
- Polynomial.eval₂proof · cited by 267
- IsAlgebraicstatement and proof · cited by 163
- Polynomial.Monic.ne_zeroproof · cited by 63
Cited by18
Results whose statement or proof uses this declaration.
- isAlgebraic_iff_isIntegralproof · cited by 12
- IsAlgebraic.of_finiteproof · cited by 8
- isAlgebraic_of_isFractionRingproof · cited by 6
- IsAlgebraic.restrictScalars_of_isIntegralproof · cited by 4
- IntermediateField.isSeparable_adjoin_simple_iff_isSeparableproof · cited by 4
- IsPrimitiveRoot.norm_pow_sub_one_of_prime_pow_ne_twoproof · cited by 4
- Ideal.comap_ne_bot_of_integral_memproof · cited by 2
- IsGalois.is_separable_splitting_fieldproof · cited by 2
- Polynomial.not_weaklyQuasiFiniteAtproof · cited by 2
- IsIntegral.trans_isAlgebraicproof · cited by 1
- Complex.isAlgebraic_cos_rat_mul_piproof · cited by 1
- Complex.isAlgebraic_sin_rat_mul_piproof · cited by 1