The adele norm as a product of conjugates #
Let L / K be a Galois extension of number fields. The norm of x ∈ L is the product of its
Galois conjugates, N_{L/K}(x) = ∏_σ σ x (Algebra.norm_eq_prod_automorphisms). This file
proves the same formula for adeles: extended back to L, the adele norm N_{L/K}(a) of an adele
a of L is the product of the Galois conjugates σ a (adeleExtension_adeleNorm), and
likewise for finite and for infinite adeles separately.
The formula is an instance of a transfer principle, which does not need L / K to be normal. Let
M be an extension of both K and L, and σ i finitely many ring automorphisms of M such
that N_{L/K}(x) = ∏ i, σ i x in M for every x ∈ L; for instance M Galois over K and the
σ i representatives of the cosets of Gal(M/L) in Gal(M/K). Then the adele norm satisfies
the same formula: extended to M, N_{L/K}(a) = ∏ i, σ i a
(adeleExtension_adeleNorm_of_algebraMap_norm_eq_prod). Both sides are continuous and
multiplicative in a and agree on the diagonal L, by hypothesis. The diagonal L is dense in
the finite adeles of L (strong approximation) and in the infinite adeles of L (weak
approximation), so the two sides agree on each factor, and on the adeles.
This is what identifies the norm map of ideles with the norm of the Galois module of ideles, and hence the norms between the levels of the idele-class formation with the norm maps of idele classes.
Main results #
TauCeti.GlobalNumberFields.finiteAdeleExtension_finiteAdeleNorm_of_algebraMap_norm_eq_prod,TauCeti.GlobalNumberFields.infiniteAdeleExtension_infiniteAdeleNorm_of_algebraMap_norm_eq_prod,TauCeti.GlobalNumberFields.adeleExtension_adeleNorm_of_algebraMap_norm_eq_prod: a product formula for the norm of field elements transfers to finite adeles, infinite adeles and adeles.TauCeti.GlobalNumberFields.finiteAdeleExtension_finiteAdeleNorm: the finite adele norm of a Galois extension, extended back toL, is the product of the Galois conjugates.TauCeti.GlobalNumberFields.infiniteAdeleExtension_infiniteAdeleNorm: the same for infinite adeles.TauCeti.GlobalNumberFields.adeleExtension_adeleNorm: the same for adeles.
References #
- J. Neukirch, Algebraic Number Theory, Chapter VI, §2.
A product formula for the norm transfers to finite adeles. Let L/K be an extension of
number fields and M an extension of both. If for some ring automorphisms σ i of M the norm
N_{L/K}(x) of every x ∈ L, read in M, is ∏ i, σ i x, then the same holds for the finite
adele norm: extended to M, the norm of a finite adele a of L is ∏ i, σ i a.
A product formula for the norm transfers to infinite adeles. If for some ring
automorphisms σ i of M the norm N_{L/K}(x) of every x ∈ L, read in M, is ∏ i, σ i x,
then extended to M, the norm of an infinite adele a of L is ∏ i, σ i a.
A product formula for the norm transfers to adeles. If for some ring automorphisms σ i
of M the norm N_{L/K}(x) of every x ∈ L, read in M, is ∏ i, σ i x, then extended to
M, the adele norm N_{L/K}(a) of an adele a of L is ∏ i, σ i a. The extension L/K need
not be normal: M may be a Galois extension of K containing L, and the σ i representatives
of the cosets of Gal(M/L) in Gal(M/K).
The finite adele norm of a Galois extension is the product of the Galois conjugates:
extended back to L, the norm of a finite adele a of L is ∏_σ σ a.
The infinite adele norm of a Galois extension is the product of the Galois conjugates:
extended back to L, the norm of an infinite adele a of L is ∏_σ σ a.
The adele norm of a Galois extension is the product of the Galois conjugates: extended
back to L, the norm of an adele a of L is ∏_σ σ a.