The explicit cyclotomic symbol kills local norms #
Let M be a finite extension of ℚ_p inside ℚ_p(μ_m) ⊆ AlgebraicClosure ℚ_[p]. Every
σ ∈ G_{ℚ_p} acting on μ_m through the explicit symbol cyclotomicSymbol m p (N_{M/ℚ_p} y) of a
norm from M fixes M. This is the local input of the cyclotomic normalization of the local Artin
map: it establishes triviality of the explicit symbol on local norms and supplies an input to the
later comparison with the Artin map, with no reciprocity law and no Lubin–Tate theory.
The proof is global. The number field K = ℚ(μ_m) ∩ M cuts out in Gal(ℚ_p(μ_m)/ℚ_p) the same
subgroup as M, so it suffices to show that σ fixes K. Weak approximation gives Y ∈ K close
to y at the embedding K ⊆ M, close to 1 at the other p-adic places and at the places above
the other primes dividing m, and totally positive. In the product formula for N_{K/ℚ}(Y), the
symbol at p is that of N_{M/ℚ_p}(y), the symbols at the other primes dividing m and at the
real place are trivial, and the symbols at the primes not dividing m fix K. So the symbol of
N_{M/ℚ_p}(y) fixes K too.
Main results #
TauCeti.cyclotomicSymbol_norm_fixes: the explicit symbol of a norm fromMfixesM.
References #
- J. S. Milne, Class Field Theory, Chapter VII, Example 8.2, for the explicit symbols and their product formula.
- J. Neukirch, Algebraic Number Theory, Chapter II, §3, for weak approximation.
The explicit symbol at p kills local norms. Let M be a finite extension of ℚ_p inside
ℚ_p(μ_m). For y ∈ Mˣ, every σ ∈ G_{ℚ_p} acting on μ_m through
cyclotomicSymbol m p (N_{M/ℚ_p} y) fixes M.