Theorems · Theorem · field theory
minpoly.eq_X_sub_C
∀ {A : Type u_1} (B : Type u_2) [inst : Field A] [inst_1 : Ring B] [inst_2 : Algebra A B] [Nontrivial B] (a : A),
minpoly A ((algebraMap A B) a) = Polynomial.X - Polynomial.C aIf B/K is a nontrivial algebra over a field, and x is an element of K,
then the minimal polynomial of algebraMap K B x is X - C x.
- Defined in
- Mathlib.FieldTheory.Minpoly.Field
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldRingAlgebraNontrivial
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Ringstatement and proof · cited by 7,463
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement · cited by 5,681
- Algebra.algebraMapstatement and proof · cited by 4,706
- Nontrivialstatement and proof · cited by 2,416
- Polynomial.Xstatement · cited by 1,639
- Polynomial.Cstatement · cited by 1,598
- minpolystatement · cited by 439
- RingHom.injectiveproof · cited by 187
Cited by6
Results whose statement or proof uses this declaration.
- isSolvable_gal_minpolyproof · cited by 2
- minpoly.zeroproof · cited by 2
- IsPurelyInseparable.elemExponent_eq_zero_of_mem_rangeproof · cited by 1
- minpoly.eq_X_sub_C'proof · cited by 0
- minpoly.oneproof · cited by 0
- Valuation.coeff_zero_minpolyproof · cited by 0