Jacques Martinet February 1st, 2013

Comment and erratum to Formes quadratiques et extensions en caractéristique 2

by Anne-Marie Bergé and Jacques Martinet,

Ann. Inst. Fourier 35,2 (1985), 57–77.

One comment. This paper was intended to consider additive discriminants attached to a basis B in a finite extension L/K in characteristic 2, not merely in K/℘K, in analogy with Kummer theory in characteristic not 2, where d_{L/K}(B) exists in K^{∗}, not merely in K^{∗}/K^{∗}^{2}. A construction only modulo ℘K can be done in a much shorter way.

Erratum.

• In Remark 4.8, page 76, we have listed the Arf invariants of polynomials of degree up to 5, indeed only reduced polynomials

X^{5} + bX^{3} + cX^{2} + dX + e

in degree five (which does not matter a great deal since in odd degrees n, we can get rid by translation of the term in X^{n−1}). In the numerator of the formula for degree 5 the term bc^{3}de has disappeared during the editing process. This error was discovered by Jean-Yves Degos when writing its PhD thesis “Classes caractéristiques de représentations galoisiennes et invariants d’algèbres étales sur un corps de caractéristique 2” (Bordeaux, October 9th, 2000); see Remark 2.2.6., p. 63. The correct formula is

Arf_{p} ≡ \frac{b^{3}c^{2}d^{2} + c^{4}d^{2} + c^{5}e + bc^{3}de + b^{5}e^{2} + b^{3}de^{2} + e^{4} + bce^{3}}{(e^{2} + bce + c^{2}d)^{2}} .

[Degos has published one paper extracted from his thesis, namely On multiplicative transfers in singular and Galois cohomologies, Math. Proc. Cambridge Philos. Soc. 136 (2004), 443–463, however not relevant for our purpose.]

• The complete reference for [12] is: J.-P. Serre, L’invariant de Witt de la forme Tr(x^{2}), Comm. Math. Helvet. 59 (1984), 651–676, = Œuvres, t. III, n^{∘} 131, pp. 675–700.