Theorems · Theorem · field theory
minpoly.exists_algEquiv_of_root
∀ {K : Type u_6} {L : Type u_7} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] [Normal K L] {x y : L},
IsAlgebraic K y → (Polynomial.aeval x) (minpoly K y) = 0 → ∃ σ, σ x = yIf x : L is a root of minpoly K y, then we can find (σ : Gal(L/K)) with σ x = y.
That is, x and y are Galois conjugates.
- Defined in
- Mathlib.FieldTheory.Normal.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 151 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
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Cites21
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
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement · cited by 5,681
- Algebra.algebraMapproof · cited by 4,706
- AlgHomstatement · cited by 3,236
- AlgEquivstatement and proof · cited by 1,681
- Polynomial.aevalstatement and proof · cited by 615
- minpolystatement and proof · cited by 439
- IntermediateField.adjoinproof · cited by 382
- IsAlgebraicstatement and proof · cited by 163
- Normalstatement and proof · cited by 92
Cited by1
Results whose statement or proof uses this declaration.
- minpoly.exists_algEquiv_of_root'proof · cited by 1