Theorems · Theorem · commutative algebra
PowerBasis.trace_gen_eq_nextCoeff_minpoly
∀ {S : Type u_2} [inst : CommRing S] {K : Type u_4} [inst_1 : Field K] [inst_2 : Algebra K S] [Nontrivial S]
(pb : PowerBasis K S), (Algebra.trace K S) pb.gen = -(minpoly K pb.gen).nextCoeffGiven pb : PowerBasis K S, the trace of pb.gen is -(minpoly K pb.gen).nextCoeff.
- Defined in
- Mathlib.RingTheory.Trace.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 127 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites26
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- RingHom.idstatement · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- LinearMapstatement · cited by 10,215
- Fieldstatement and proof · cited by 7,404
- Polynomialproof · cited by 5,681
- Nontrivialstatement and proof · cited by 2,416
- Fintype.cardproof · cited by 1,386
- Polynomial.natDegreeproof · cited by 1,105
- Polynomial.coeffproof · cited by 1,045
- minpolystatement and proof · cited by 439
Cited by2
Results whose statement or proof uses this declaration.
- PowerBasis.trace_gen_eq_sum_rootsproof · cited by 2
- trace_adjoinSimpleGenproof · cited by 2