Theorems · Theorem · number theory
NumberField.FinitePlace.prod_eq_inv_abs_norm
∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K] {x : K},
x ≠ 0 → ∏ᶠ (w : NumberField.FinitePlace K), w x = ↑|(Algebra.norm ℚ) x|⁻¹For any non-zero x in K, the product of w x, where w runs over FinitePlace K, is
equal to the inverse of the absolute value of Algebra.norm ℚ x.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldNumberField
Around this declaration
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Cites34
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
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- absstatement and proof · cited by 1,814
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- map_mulproof · cited by 1,137
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- nonZeroDivisorsproof · cited by 895
- NumberFieldstatement and proof · cited by 653
Cited by1
Results whose statement or proof uses this declaration.
- NumberField.prod_abs_eq_oneproof · cited by 0