Theorems · Theorem · number theory
NumberField.InfinitePlace.prod_eq_abs_norm
∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K] (x : K), ∏ w, w x ^ w.mult = ↑|(Algebra.norm ℚ) x|The infinite part of the product formula : for x ∈ K, we have Π_w ‖x‖_w = |norm(x)| where
‖·‖_w is the normalized absolute value for w.
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 296 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldNumberField
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites34
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
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- RingHomproof · cited by 10,189
- Fieldstatement and proof · cited by 7,404
- Complexproof · cited by 5,565
- Norm.normproof · cited by 5,413
- Algebra.algebraMapproof · cited by 4,706
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- Finset.univstatement and proof · cited by 3,473
- AlgHomproof · cited by 3,236
- Finset.prodstatement and proof · cited by 2,356
- absstatement and proof · cited by 1,814
Cited by8
Results whose statement or proof uses this declaration.
- NumberField.mixedEmbedding.norm_eq_normproof · cited by 7
- NumberField.Units.dirichletUnitTheorem.seq_nextproof · cited by 3
- NumberField.InfinitePlace.one_le_of_lt_oneproof · cited by 2
- NumberField.Units.sum_mult_mul_logproof · cited by 2
- NumberField.mixedEmbedding.fundamentalCone.prod_expMapBasis_powproof · cited by 1
- NumberField.Units.abs_det_eq_abs_detproof · cited by 1
- NumberField.InfiniteAdeleRing.coe_norm_eq_abs_normproof · cited by 0
- NumberField.prod_abs_eq_oneproof · cited by 0