CORRECTIONS to the PAPERS [7] and [8] on QUATERNION EXTENSIONS

Jacques Martinet First edited on November 7th, 2014; revised on November 18th, 2014.

In a letter dated from March 13th, 1996, Jean Cougnard pointed out to me that the sentence in Theorem IV-1 of [7]:

“On peut prendre pour φ n’importe lequel des éléments

\frac{±1 ± \sqrt{d_{1}} ± \sqrt{d_{2}} ± \sqrt{d_{3}}}{4} de A′ ”

is not correct. Actually, once such an element φ has been found, the other possible choices are deduced from φ by an even number of change of signs.

Probably the same year Cougnard had also noticed that I had not correctly reproduced in [8] Leopoldt’s basis for the orders associated to biquadratic extensions.

These two questions have nothing to do with quaternion extensions of degree 8: they are related to properties of biquadratic fields, and whether such a field can be embedded in a quaternion extension does not matter. I give below the correct answers, based on elementary calculations, however using a more symmetric notation.

Let K/ℚ be a biquadratic extension, with quadratic subfields k_{1}, k_{2}, k_{3}. Write k_{i} = ℚ(θ_{i}) where m_{i} := θ_{i}^{2} is a square-free integer and denote by d_{i} the discriminant of k_{i} and by D that of K; thus we have d_{i} = m_{i} if m_{i} ≡ 1 mod 4 and d_{i} = 4m_{i} if m_{i} ≡ 2 or 3 mod 4, and D = d_{1}d_{2}d_{3}.

We may and shall choose the square roots θ_{i} of the m_{i} in such a way that θ_{1}θ_{1}θ_{3} be the positive square root of m_{1}m_{2}m_{3}.

[In what follows such a choice is important only when K/ℚ is unramified at 2. In this case this solves the problem raised for [7].]

Set moreover ω_{i} = \frac{1+θ_{i}}{2} , ω_{i,j} = \frac{θ_{i}+θ_{j}}{2} , and ω = \frac{1+θ_{1}+θ_{2}+θ_{3}}{4} . Also let g = {1, s_{1}, s_{2}, s_{3}} be the Galois group of K/ℚ where s_{i} is the identity on k_{i}, and denote by e the ramification index of 2 (e = 1, 2 or 4 ; e = 1 if and only if K/ℚ is tamely ramified). If e = 2 (resp. e = 4) we may index the fields k_{i} so as to have m_{1} ≡ 1 mod 4 (resp. m_{1} ≡ 3 mod 4). Denote by ℤ_{K} the ring of integers of K and by 𝔒_{K} the order associated with ℤ_{K} in ℚ[g], namely

𝔒_{K} = {λ ∈ ℚ[g] | λℤ_{K} ⊂ ℤ_{K}}.

With these conventions, explicit calculations will show that basis elements over ℤ for ℤ_{K} and for 𝔒_{K} may be chosen as follows:

• \underline{e = 1}. Z_{K} : 1, ω_{1}, ω_{2}, ω; 𝔒_{K} : 1, s_{1}, s_{2}, s_{3} .

• \underline{e = 2}. Z_{K} : 1, ω_{1}, θ_{2}, ω_{2,3} ; 𝔒_{K} : 1, \frac{1+s_{1}}{2} , s_{2}, \frac{s_{2}+s_{3}}{2} .

• \underline{e = 4}. Z_{K} : 1, θ_{1}, θ_{2}, ω_{2,3} ; 𝔒_{K} : 1, \frac{1+s_{1}}{2} , \frac{1+s_{2}}{2} , \frac{1+s_{1}+s_{2}+s_{3}}{4} .

[Calculating directly D provides a check for ℤ_{K} , index calculations for 𝔒_{K}. Note that the equality 𝔒_{K} = ℤ[g] for tame extensions is a general result valid relatively to any Dedekind domain of characteristic zero.]

A remark on cases A—E in Proposition 1 of [8].

Somebody (who? when?) pointed out to me that case C does not occur in Fröhlich’s paper [Fr] below. Case C indeed does not exist. This imperfection in [8] will be corrected in proofs, a complement to [8] I am now writing.

[Fr] A. Fröhlich, Artin Root Numbers and Normal Integral Bases for Quaternion Fields, Invent. Math 17 (1972), 143–166.