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feat(NumberTheory/Height/NumberField): the absolute height of an algebraic number #41605

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feat(NumberTheory/Height/NumberField): the absolute height of an algebraic number #41605

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tb65536 Jul 10, 2026
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better docstring
tb65536 Jul 10, 2026
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10 changes: 10 additions & 0 deletions Mathlib/NumberTheory/Height/NumberField.lean
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Expand Up @@ -128,6 +128,16 @@ lemma mulHeight_eq {ι : Type*} {x : ι → K} (hx : x ≠ 0) :
simp only [FinitePlace.coe_apply, InfinitePlace.coe_apply, Height.mulHeight_eq hx,
prod_archAbsVal_eq, prod_nonarchAbsVal_eq fun v ↦ ⨆ i, v (x i)]

open Classical IntermediateField in
/-- The absolute multiplicative height of an algebraic number. This is defined for elements of any
field of characteristic zero, with a junk value of `0` if the element is not algebraic. -/
noncomputable def absMulHeight₁ {K : Type*} [Field K] [CharZero K] (x : K) : ℝ :=
if hx : IsIntegral ℚ x then
haveI : FiniteDimensional ℚ ℚ⟮x⟯ := adjoin.finiteDimensional hx
haveI : NumberField ℚ⟮x⟯ := {}
(Height.mulHeight₁ (AdjoinSimple.gen ℚ x)) ^ (Module.finrank ℚ ℚ⟮x⟯ : ℝ)⁻¹
else 0

variable (K) in
lemma totalWeight_eq_sum_mult : totalWeight K = ∑ v : InfinitePlace K, v.mult := by
simp only [totalWeight]
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