The decomposition law above 2 in a multiquadratic field #
Let K = ℚ(√d₁, …, √dₙ) be a number field generated over ℚ by square roots r i of integers
d i not divisible by 4 (for instance squarefree), and let Q be a prime of 𝓞 K above 2.
When every dᵢ is 1 modulo 4 the prime 2 is unramified
(TauCeti.Multiquadratic.inertia_eq_bot_of_forall_mod_four_eq_one). This file computes the
ramification index e and the residue degree f of Q in general. The number of primes above
2 is computed in TauCeti.NumberTheory.Multiquadratic.Dyadic.Decomposition. Unlike at an odd
prime, e can be 4:
e = 4exactly when somedᵢis3modulo4and somedⱼis even, or somedᵢis2and somedⱼis6modulo8;- otherwise
e = 2if somedᵢis not1modulo4, ande = 1if everydᵢis.
Behind the case split are the square classes of ℚ₂. Modulo squares and the unramified class of
5, an integer not divisible by 4 falls into one of four classes, according to whether it is
1 or 3 modulo 4 when odd, and 2 or 6 modulo 8 when even; these classes form a group of
order 4, and e is the order of the subgroup generated by the classes of the dᵢ. Likewise
f = 2 exactly when the subgroup of the square classes of ℚ₂ generated by the dᵢ contains the
class of 5: some subset product ∏_{i ∈ T} dᵢ is a rational square times an integer that is 5
modulo 8. This criterion needs no hypothesis on the radicands. The proofs below do not use ℚ₂,
and work with the action of inertia and Frobenius on square roots in K instead.
- Upper bound. An inertia element
τfixes every square root of an integer that is1modulo4(TauCeti.Multiquadratic.apply_eq_self_of_mem_inertia_of_mod_four_eq_one). Applied torᵢ rⱼ,rᵢ rⱼ / 2andrᵢ rⱼ rₖ / 2for suitable radicands, this shows thatτis determined by its signs on at most two of therᵢ, so the inertia group has at most four elements, and at most two outside the first case above. - Lower bound. A square root
xof an integer that is2or3modulo4generates a quadratic field ramified at2, so some inertia element negatesx(TauCeti.Multiquadratic.exists_mem_inertia_apply_eq_neg_of_mod_four_eq_two_or_three, fromTauCeti.NumberField.exists_mem_inertia_apply_eq_neg). In the first case above there are two such rootsy,zwhose product, up to a factor2, is a third. A group of exponent2with at most four elements that negates each ofy,zandy zhas exactly four elements. - Residue degree.
f = 1exactly when a FrobeniusφatQlies in the inertia group. A square root of an integer5modulo8is fixed by inertia and negated byφ. Conversely, ifφis not in the inertia group, an index-two subgroup containing inertia but notφcuts out a quadratic field generated by such a square root. Square-class descent then rewrites the criterion in terms of the radicands.
Since K / ℚ is Galois, the number g of primes above 2 then satisfies g · e · f = [K : ℚ].
When the dᵢ are square-class independent (no nonempty subset product is a square),
[K : ℚ] = 2 ^ n, so the formulas in
TauCeti.NumberTheory.Multiquadratic.Dyadic.Decomposition give g explicitly and complete the
decomposition law above 2 in that case.
Main results #
TauCeti.Multiquadratic.exists_mem_inertia_apply_eq_neg_of_mod_four_eq_two_or_three: every prime above2has an inertia element negating a given square root of an integer that is2or3modulo4.TauCeti.Multiquadratic.card_inertia_le_fourandTauCeti.Multiquadratic.ramificationIdx_le_four: the inertia group above2has at most four elements.TauCeti.Multiquadratic.ramificationIdx_eq_four_iff: the criterion fore = 4above.TauCeti.Multiquadratic.inertia_eq_bot_iff_forall_mod_four_eq_one: the criterion fore = 1.TauCeti.Multiquadratic.ramificationIdx_eq_two_iff_of_liesOver_two: the criterion fore = 2above.TauCeti.Multiquadratic.inertiaDeg_eq_two_iff_exists_sq_eq_mod_eight_eq_fiveand itsinertiaDeg_eq_one_iff_not_exists_sq_eq_mod_eight_eq_fivecompanion: for any GaloisK / ℚof exponent two, a prime above2has residue degree2exactly whenKcontains a square root of an integer that is5modulo8.TauCeti.Multiquadratic.inertiaDeg_eq_two_iff_exists_prod_eq_mod_eight_eq_five_mul_sqandTauCeti.Multiquadratic.inertiaDeg_eq_one_iff_not_exists_prod_eq_mod_eight_eq_five_mul_sq: the criteria forf = 2andf = 1above, in terms of the radicands.TauCeti.Multiquadratic.inertiaDeg_eq_two_iff_exists_squarefree_prod_eq_mod_eight_eq_five_mul_sq: the criterion in terms of integer squarefree parts of subset products.TauCeti.Multiquadratic.inertiaDeg_eq_two_iff_exists_prod_mod_eight_eq_five_of_not_four_dvd_prodandinertiaDeg_eq_one_iff_forall_prod_mod_eight_ne_five_of_not_four_dvd_prod: when no subset product is divisible by four, finite tests on the subset products modulo eight.
References #
- D. A. Cox, Primes of the Form x² + ny², §5.B.
- J. Neukirch, Algebraic Number Theory, Chapter I, §9 and Chapter II, §5.
Inertia elements negating a ramified square root #
Inertia above 2 negates the square roots of integers that are 2 or 3 modulo 4. Let
K / ℚ be Galois, let x ∈ K square to an integer c ≡ 2 or 3 (mod 4), and let Q be a prime
of 𝓞 K above 2. Then some element τ of the inertia group of Q satisfies τ x = -x.
Inertia elements fixing products of roots #
Counting inertia elements #
The ramification index above 2 #
The inertia group above 2 has at most four elements. Let K be generated over ℚ by
square roots r i of integers d i not divisible by 4, and let Q be a prime of 𝓞 K above
2. Then the inertia group of Q in Gal(K/ℚ) has at most four elements, however many radicands
there are.
The ramification index above 2 is at most 4. Let K be generated over ℚ by square
roots of integers not divisible by 4, and let Q be a prime of 𝓞 K above 2. Then the
ramification index of Q over 2 is at most 4.
Ramification index 4 above 2. Let K be generated over ℚ by square roots of
integers d i not divisible by 4, and let Q be a prime of 𝓞 K above 2. Then the
ramification index of Q over 2 is 4 exactly when some d i is 3 modulo 4 and some
d j is even, or some d i is 2 and some d j is 6 modulo 8.
Trivial inertia above 2. Let K be generated over ℚ by square roots of integers
d i not divisible by 4, and let Q be a prime of 𝓞 K above 2. Then the inertia group of
Q is trivial exactly when every d i is 1 modulo 4.
Ramification index 2 above 2. Let K be generated over ℚ by square roots of
integers d i not divisible by 4, and let Q be a prime of 𝓞 K above 2. Then the
ramification index of Q over 2 is 2 exactly when some d i is not 1 modulo 4 (so
that 2 ramifies) but the criterion of TauCeti.Multiquadratic.ramificationIdx_eq_four_iff
fails.
The residue degree above 2 #
The Frobenius φ at a prime Q above 2 acts on a square root of an integer c ≡ 1 (mod 4)
trivially exactly when c ≡ 1 (mod 8) (TauCeti.isArithFrobAt_apply_sqrt_eq_self_iff_mod_eight),
whether or not 2 ramifies, while the inertia group fixes every such square root. The residue
degree is 1 exactly when φ lies in the inertia group, so a square root of an integer
c ≡ 5 (mod 8) forces residue degree 2. Conversely, if φ is not in the inertia group, an
index-two subgroup containing the inertia group and avoiding φ cuts out a quadratic field
ℚ(y) with y ^ 2 a squarefree integer s; inertia fixes y, so s ≡ 1 (mod 4), and φ moves
y, so s ≡ 5 (mod 8).
Residue degree two above 2, intrinsically. Let K / ℚ be Galois with group of exponent
dividing two, and let Q be a prime of 𝓞 K above 2, ramified or not. Then Q has residue
degree 2 exactly when K contains a square root of an integer that is 5 modulo 8.
In an exponent-two Galois number field, a prime above two has residue degree one exactly when the field contains no square root of an integer congruent to five modulo eight.
Residue degree two above 2. Let K be generated over ℚ by square roots r i of
integers d i, and let Q be a prime of 𝓞 K above 2, ramified or not. Then Q has residue
degree 2 exactly when some subset product ∏_{i ∈ T} dᵢ lies in the rational square class of an
integer that is 5 modulo 8.
Residue degree one above 2. Let K be generated over ℚ by square roots r i of
integers d i, and let Q be a prime of 𝓞 K above 2, ramified or not. Then Q has residue
degree 1 exactly when no subset product ∏_{i ∈ T} dᵢ lies in the rational square class of an
integer that is 5 modulo 8.
The residue degree above two is two exactly when the squarefree part of some subset product is congruent to five modulo eight. All factors in this criterion are integers.
When no subset product of the integer radicands is divisible by four, the residue degree above two is two exactly when one of the finitely many subset products is congruent to five modulo eight.
When no subset product of the integer radicands is divisible by four, the residue degree above two is one exactly when every subset product is not congruent to five modulo eight.