Theorems · Theorem · commutative algebra
Ideal.IsField.of_isPrincipalIdealRing_polynomial
∀ {R : Type u_1} [inst : CommRing R] [IsDomain R] [IsPrincipalIdealRing (Polynomial R)], IsField RGiven a domain R, if R[X] is a principal ideal ring, then R is a field.
- Defined in
- Mathlib.RingTheory.Polynomial.Quotient
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 126 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Polynomialstatement and proof · cited by 5,681
- HasQuotient.Quotientproof · cited by 2,301
- IsDomainstatement and proof · cited by 2,196
- Polynomial.Xproof · cited by 1,639
- Polynomial.Cproof · cited by 1,598
- Ideal.spanproof · cited by 948
- AlgEquiv.symmproof · cited by 615
- IsPrincipalIdealRingstatement and proof · cited by 131
- IsFieldstatement and proof · cited by 103
- MulEquiv.isFieldproof · cited by 14
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