Biquaternion algebras and quartic extensions

Abstract

Biquaternion algebras and quartic extensions Publications mathématiques de l'I.H.É.S., tome 77 (1993), p. 63-102 <http://www.numdam.org/item?id=PMIHES_1993__77__63_0> © Publications mathématiques de l'I.H.É.S., 1993, tous droits réservés. L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ BIQUATERNION ALGEBRAS AND QUARTIC EXTENSIONS by TSIT YUEN LAM (1), DAVID B. LEEP (2), JEAN-PIERRE TIGNOL (3)

Cohomological operations for the Brauer group and Witt kernels of a quartic field extension

“…On the other hand, for any α ∈ 2 Br(L/F ), there exist c, e ∈ F * such that α = (a, c)+(b, e) ( [6], Theorem 3.13), so in this case the statement is obvious. Now assume that L/F is not biquadratic.…”

“…On the other hand, it has been proven in ( [6], Theorem 3.13) that the ideal W (L/F ) is generated by Pfister forms ux 2 + vx + w 2 , x and u, t , where t, x, y ∈ F * . It easily follows that any element of 2 Br(L/F ) is equal either to…”

Cohomology and graded Witt group kernels for extensions of degree four in characteristic two

“…The condition on f, t ∈ F that t = ℘(f ) + s is an analogue of the generating polynomials that Lam, Leep and Tignol found for Witt kernels of quartic extensions away from characteristic two [8]. Their result considered the two-fold Pfister forms C := { e, −P (e) | e ∈ F * } where P (e) = ae 2 + be + c 2 and the quartic extension was F ( √ a, b + 2c √ a).…”

The Witt ring kernel for a fourth degree field extension and related problems

“…In particular, in terminology of [1] the ideal W (L/F ) ∩ I n (F ) is an n-Pfister ideal, i.e. is generated by n-fold Pfister forms.…”