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Formes quadratiques et extensions en caractéristique 2. (Quadratic forms and extensions of charactéristic two). (French) Zbl 0539.10018

Let \(E/K\) be a finite separable extension. Using permutation groups, one can attach to \(E/K\) a quadratic extension \(\tilde E\) of \(K\) (or \(\tilde E=K)\). When the characteristic of \(K\) is not 2, \(\tilde E\) corresponds to an element of \(K^*/K^{*2}\), which is the discriminant \(d_{E/K}\) of the quadratic form \(x\mapsto Tr_{E/K}(x^ 2)\). When \(K\) is of characteristic 2, we use the quadratic form \(x\mapsto T_ 2(x)\) (the coefficient of \(X^{n-2}\) in the characteristic polynomial of \(x\) in \(E\) or \(E\times K)\) to define an additive discriminant \(d^+_{E/K}\in K/{\mathcal P}(K)\) which plays the same role. More precisely, we define \(d^+_{E/K}(B)\) for a given basis of \(E/K\) by lifting in characteristic 0. We find the relation between \(d^+\) and the Arf invariant, solving a conjecture of Ph. Revoy (which has also been solved independently by A. R. Wadsworth [Linear Multilinear Algebra 17, 235-263 (1985; Zbl 0567.12017).
We then prove that the quadratic space \((E,T_ 2)\) or \((E\times K,T_ 2)\) (the dimension must be even) is a direct sum of hyperbolic planes and of one plane with trivial Clifford invariant which defines the Arf invariant. (The situation is quite different in the case of characteristic \(\neq 2\) [cf. J.-P. Serre, Comment. Math. Helv. 59, 651-676 (1984; Zbl 0565.12014)]. Finally, the results are applied to the reduction of equations ”à la Klein”. A typical result is the following one: an extension E/K of degree five can be defined by some polynomial \(X^ 5+aX+b\) (or \(X^ 5+tX+t)\) of \(K[X]\) if and only if its Arf invariant is trivial.

MSC:

11E04 Quadratic forms over general fields
11E16 General binary quadratic forms
11R23 Iwasawa theory
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References:

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