Theorems · Theorem · field theory
Algebra.transcendental_iff_not_isAlgebraic
∀ {R : Type u} {A : Type v} [inst : CommRing R] [inst_1 : Ring A] [inst_2 : Algebra R A],
Algebra.Transcendental R A ↔ ¬Algebra.IsAlgebraic R A- Defined in
- Mathlib.RingTheory.Algebraic.Defs
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 118 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- Algebra.IsAlgebraicstatement · cited by 322
- IsAlgebraicproof · cited by 163
- Algebra.Transcendentalstatement · cited by 17
Cited by8
Results whose statement or proof uses this declaration.
- RatFunc.transcendental_of_ne_Cproof · cited by 3
- trdeg_eq_zero_iffproof · cited by 2
- IsTranscendenceBasis.nonempty_iff_transcendentalproof · cited by 1
- RatFunc.finrank_eq_max_natDegreeproof · cited by 1
- Algebra.injective_of_transcendentalproof · cited by 1
- Algebra.Transcendental.ringHom_of_comp_eqproof · cited by 1
- trdeg_ne_zero_iffproof · cited by 0
- Algebra.Transcendental.of_ringHom_of_comp_eqproof · cited by 0