Hilbert sign prescription #
Let K be a number field. Hilbert sign prescription (O'Meara 71:19a) says: if S is a finite
set of finite places and T a finite set of real places of K with #S + #T even, and b ∈ Kˣ
is a nonsquare in K_v for every v ∈ S and negative at every w ∈ T, then some a ∈ Kˣ has
(a_v, b_v)_v = -1 for v ∈ S ∪ T, (a_v, b_v)_v = 1 at every other finite and real place.
Hilbert's product formula only says that the sign pattern of a given pair has even support; it
does not construct a pair with a prescribed pattern. The existence comes from the idele Hilbert
character φ_b(x) = ∏_v (x_v, b)_v and the norm index theorem. For a number field L ⊇ K in which
b is a square, φ_b vanishes on the principal ideles by the product formula, and on the idele
norms N_{L/K}(𝕀_L) because every coordinate of an idele norm is a local norm. If b is a
nonsquare somewhere then ker φ_b has index two, and if N_{L/K}(C_L) has index dividing two in
C_K, as the norm index theorem [C_K : N_{L/K}(C_L)] = [L : K] gives for L = K(√b), then
ker φ_b = Kˣ · N_{L/K}(𝕀_L).
An idele that is a local non-norm exactly at the places of S ∪ T lies in ker φ_b, because
#S + #T is even, so it is a · N(y) for some a ∈ Kˣ; this a has the prescribed symbols.
This file proves the kernel description and sign prescription from Hilbert reciprocity
(hilbertSymbol_productFormula) and the index bound for one L. The index bound is the Kummer
case of the second fundamental inequality, the bound [C_F : N_{L/F}(C_L)] ≤ p for Galois
extensions L/F of prime degree p with μ_p ⊆ F: at p = 2, together with the first
inequality, it gives the norm index of K(√b)/K in Type, and Shrink.ringEquiv K transports
sign prescription to number fields in any universe. Finally, choosing b by weak approximation
gives the version in which b is not prescribed (O'Meara 71:19).
Main results #
TauCeti.NumberField.QuadraticForm.sup_range_ideleNormMap_eq_ker_ideleHilbertCharacter:Kˣ · N_{L/K}(𝕀_L) = ker φ_b, from the index bound.TauCeti.NumberField.QuadraticForm.exists_hilbertSymbol_eq_ite_of_index_dvd_two: sign prescription from the index bound.TauCeti.NumberField.QuadraticForm.exists_hilbertSymbol_eq_ite: sign prescription over every number field.TauCeti.NumberField.QuadraticForm.exists_pair_hilbertSymbol_eq_ite: a paira, bwith prescribed symbols at an even number of places.
References #
- O. T. O'Meara, Introduction to Quadratic Forms, Springer (1963), 71:19 and 71:19a, with 65:21 for the norm index.
- J. S. Milne, Class Field Theory, v4.03, Chapter VII, §5 and §6.
The kernel of the idele Hilbert character. Let b ∈ Kˣ be a nonsquare at some finite place
or negative at some real place, and let L be a number field over K in which b is a square.
Suppose that the idele-class norm group N_{L/K}(C_L) has index dividing two in C_K, as the norm
index theorem gives for L = K(√b). Then the kernel of φ_b is Kˣ · N_{L/K}(𝕀_L), the
subgroup of 𝕀_K generated by the principal ideles and the idele norms.
Hilbert sign prescription from the norm index. Let L be a number field over K in which
b ∈ Kˣ is a square, such that N_{L/K}(C_L) has index dividing two in C_K. Let S be a finite
set of finite places at which b is a nonsquare and T a finite set of real places at which b is
negative, with #S + #T even. Then some a ∈ Kˣ has Hilbert symbol (a, b)_v = -1 at the places
of S and of T, and 1 at every other finite and real place.
Sign prescription over a number field #
Hilbert sign prescription (O'Meara 71:19a). Let S be a finite set of finite places at
which b ∈ Kˣ is a nonsquare and T a finite set of real places at which b is negative, with
#S + #T even. Then some a ∈ Kˣ has Hilbert symbol (a, b)_v = -1 at the places of S and of
T, and 1 at every other finite and real place.
Hilbert sign prescription with a free second argument (O'Meara 71:19). For a finite set
S of finite places and a finite set T of real places with #S + #T even, some a, b ∈ Kˣ have
Hilbert symbol (a, b)_v = -1 at the places of S and of T, and 1 at every other finite and
real place.