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module
public import Mathlib.NumberTheory.NumberField.Basic
public import FormalConjecturesForMathlib.Algebra.QuadraticAlgebra.Instancespublic sectionopen Algebra (IsQuadraticExtension)open Module NumberField
Any QuadraticAlgebra ℚ a b that is a field is automatically a quadratic extension
of ℚ, i.e., a degree-2 extension. Combined with IsQuadraticField.instNumberField,
this gives NumberField (QuadraticAlgebra ℚ a b) for free.
instance QuadraticAlgebra.instIsQuadraticExtension (a b : ℚ) [Fact (∀ r : ℚ, r ^ 2 ≠ a + b * r)] :
IsQuadraticExtension ℚ (QuadraticAlgebra ℚ a b) where
finrank_eq_two' := QuadraticAlgebra.finrank_eq_two a b
A quadratic field is a number field: it has characteristic zero
and is finite-dimensional over ℚ.
instance Algebra.IsQuadraticExtension.to_numberField (K : Type*) [Field K] [CharZero K]
[IsQuadraticExtension ℚ K] : NumberField K where
to_finiteDimensional := .of_finrank_pos <| K:Type u_1inst✝²:Field Kinst✝¹:CharZero Kinst✝:IsQuadraticExtension ℚ K⊢ 0 < finrank ℚ K All goals completed! 🐙