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feat(number_theory/number_field/norm): add file (#17672)
We add `norm' : (π L) β* (π K)`, that is `algebra.norm K` as a morphism between the rings of integers. From flt-regular
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/- | ||
Copyright (c) 2022 Riccardo Brasca. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Riccardo Brasca, Eric Rodriguez | ||
-/ | ||
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import number_theory.number_field.basic | ||
import ring_theory.norm | ||
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/-! | ||
# Norm in number fields | ||
Given a finite extension of number fields, we define the norm morphism as a function between the | ||
rings of integers. | ||
## Main definitions | ||
* `ring_of_integers.norm K` : `algebra.norm` as a morphism `(π L) β* (π K)`. | ||
## Main results | ||
* `algebra.dvd_norm` : if `L/K` is a finite Galois extension of fields, then, for all `(x : π L)` | ||
we have that `x β£ algebra_map (π K) (π L) (norm K x)`. | ||
-/ | ||
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open_locale number_field big_operators | ||
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open finset number_field algebra | ||
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namespace ring_of_integers | ||
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variables {L : Type*} (K : Type*) [field K] [field L] [algebra K L] [finite_dimensional K L] | ||
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/-- `algebra.norm` as a morphism betwen the rings of integers. -/ | ||
@[simps] noncomputable def norm [is_separable K L] : (π L) β* (π K) := | ||
((algebra.norm K).restrict (π L)).cod_restrict (π K) (Ξ» x, is_integral_norm K x.2) | ||
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local attribute [instance] number_field.ring_of_integers_algebra | ||
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lemma coe_algebra_map_norm [is_separable K L] (x : π L) : | ||
(algebra_map (π K) (π L) (norm K x) : L) = algebra_map K L (algebra.norm K (x : L)) := rfl | ||
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lemma is_unit_norm [is_galois K L] {x : π L} : | ||
is_unit (norm K x) β is_unit x := | ||
begin | ||
classical, | ||
refine β¨Ξ» hx, _, is_unit.map _β©, | ||
replace hx : is_unit (algebra_map (π K) (π L) $ norm K x) := hx.map (algebra_map (π K) $ π L), | ||
refine @is_unit_of_mul_is_unit_right (π L) _ | ||
β¨(univ \ { alg_equiv.refl }).prod (Ξ» (Ο : L ββ[K] L), Ο x), | ||
prod_mem (Ξ» Ο hΟ, map_is_integral (Ο : L β+* L).to_int_alg_hom x.2)β© _ _, | ||
convert hx using 1, | ||
ext, | ||
push_cast, | ||
convert_to (univ \ { alg_equiv.refl }).prod (Ξ» (Ο : L ββ[K] L), Ο x) * (β (Ο : L ββ[K] L) in | ||
{alg_equiv.refl}, Ο (x : L)) = _, | ||
{ rw [prod_singleton, alg_equiv.coe_refl, id] }, | ||
{ rw [prod_sdiff $ subset_univ _, βnorm_eq_prod_automorphisms, coe_algebra_map_norm] } | ||
end | ||
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/-- If `L/K` is a finite Galois extension of fields, then, for all `(x : π L)` we have that | ||
`x β£ algebra_map (π K) (π L) (norm K x)`. -/ | ||
lemma dvd_norm [is_galois K L] (x : π L) : x β£ algebra_map (π K) (π L) (norm K x) := | ||
begin | ||
classical, | ||
have hint : (β (Ο : L ββ[K] L) in univ.erase alg_equiv.refl, Ο x) β π L := | ||
subalgebra.prod_mem _ (Ξ» Ο hΟ, (mem_ring_of_integers _ _).2 | ||
(map_is_integral Ο (ring_of_integers.is_integral_coe x))), | ||
refine β¨β¨_, hintβ©, subtype.ext _β©, | ||
rw [coe_algebra_map_norm K x, norm_eq_prod_automorphisms], | ||
simp [β finset.mul_prod_erase _ _ (mem_univ alg_equiv.refl)] | ||
end | ||
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end ring_of_integers |
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