Theorems · Theorem · field theory
IntermediateField.adjoin.finrank
∀ {K : Type u} [inst : Field K] {L : Type u_3} [inst_1 : Field L] [inst_2 : Algebra K L] {x : L},
IsIntegral K x → Module.finrank K ↥K⟮x⟯ = (minpoly K x).natDegreeIf x is an algebraic element of field K, then its minimal polynomial has degree
[K(x) : K].
- Cited by
- 14 results in Mathlib
- Foundations
- Depth 133 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- Module.finrankstatement · cited by 1,770
- Polynomial.natDegreestatement and proof · cited by 1,105
- IntermediateFieldstatement · cited by 988
- minpolystatement and proof · cited by 439
- IsIntegralstatement and proof · cited by 427
- IntermediateField.adjoinstatement · cited by 382
- PowerBasis.dimproof · cited by 74
- IntermediateField.adjoin.powerBasisproof · cited by 17
- PowerBasis.finrankproof · cited by 14
Cited by14
Results whose statement or proof uses this declaration.
- IntermediateField.finSepDegree_adjoin_simple_eq_finrank_iffproof · cited by 4
- Field.primitive_element_iff_minpoly_natDegree_eqproof · cited by 2
- IsPurelyInseparable.finrank_eq_powproof · cited by 2
- RatFunc.finrank_eq_max_natDegreeproof · cited by 1
- Algebra.normalizedTrace_minpolyproof · cited by 1
- IsGalois.IntermediateField.AdjoinSimple.card_aut_eq_finrankproof · cited by 1
- irreducible_X_pow_sub_C_of_root_adjoin_eq_topproof · cited by 1
- IntermediateField.adjoin_minpoly_coeff_of_exists_primitive_elementproof · cited by 1
- Polynomial.Gal.prime_degree_dvd_cardproof · cited by 1
- minpoly.iterateFrobenius_of_isSeparableproof · cited by 1
- Polynomial.irreducible_compproof · cited by 1
- minpoly.degree_dvdproof · cited by 1