Subfields of PadicAlgCl p and their norms #
The algebraic closure PadicAlgCl p of ℚ_p carries the spectral norm, which extends the norm of
ℚ_p and is preserved by G_{ℚ_p}. This file compares a number field K ⊆ PadicAlgCl p with the
p-adic field it generates.
- A
ℚ-subfieldKis dense in itsℚ_p-span. - For
K / ℚfinite Galois, aℚ-embedding ofKintoPadicAlgCl ppreserving the norm agrees onKwith an element ofG_{ℚ_p}: the embeddings ofKinducing the given absolute value are the conjugates underG_{ℚ_p}. These are the embeddings whose product computes the norm down toℚ_p. - The norm
N_{K/ℚ}(Y)is close toN_{M/ℚ_p}(y), forMthe fieldKgenerates overℚ_p, whenYis close toyand every embedding ofKnot coming fromG_{ℚ_p}sendsYclose to1; andN_{K/ℚ}(Y)is close to1inℚ_qwhen every embedding intoPadicAlgCl qsendsYclose to1.
Main results #
AlgEquiv.norm_apply: elements ofG_{ℚ_p}preserve the norm ofPadicAlgCl p.TauCeti.exists_mem_norm_sub_lt_of_mem_adjoinTauCeti.exists_algEquiv_apply_eq_of_norm_apply_eqTauCeti.algebraMap_norm_eq_prod_embeddings_of_le_adjoin: the norm down toℚ_pof an element ofKis the product over the embeddings ofKthat extend toG_{ℚ_p}.TauCeti.norm_algebraNorm_sub_one_ltTauCeti.norm_algebraNorm_sub_algebraNorm_ltTauCeti.place_ratCast_padicAlgCl: an embedding intoPadicAlgCl qrestricts to theq-adic absolute value onℚ.
ℚ_[p]-automorphisms of AlgebraicClosure ℚ_[p] preserve its (spectral) norm.
Let K ⊆ M with M contained in the field K generates over ℚ_p. The norm from M to
ℚ_p of an element of K is the product of its images under the ℚ-embeddings of K that
extend to elements of G_{ℚ_p}.
The global norm of Y is close to the local norm of y when Y is close to y and every
embedding of K that does not extend to a ℚ_[p]-automorphism sends Y close to 1.
If every embedding of K into PadicAlgCl q sends Y within ε ≤ 1 of 1, then the norm of
Y down to ℚ is within ε of 1 in ℚ_[q].
An element of a ℚ-subfield K of PadicAlgCl p lying in ℚ_p is fixed by every
ℚ-embedding of K into PadicAlgCl p that preserves the norm: ℚ is dense in ℚ_p.
Embeddings with the same absolute value are conjugate under G_{ℚ_p}. Let
K ⊆ PadicAlgCl p be finite Galois over ℚ. A ℚ-embedding of K into PadicAlgCl p preserving
the norm agrees on K with an element of G_{ℚ_p}, so these are exactly the embeddings
contributing to the norm down to ℚ_p in algebraMap_norm_eq_prod_embeddings_of_le_adjoin.
A ℚ-subfield of PadicAlgCl p is dense in the field it generates over ℚ_p.