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

Bergé, A.-M.; Martinet, J.

Ann. Inst. Fourier 35, No. 2, 57-77 (1985).

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

Keywords:

quadratic extension; discriminant; quadratic form; Arf invariant; quadratic space; direct sum of hyperbolic planes; plane with trivial Clifford invariant; reduction of equations

Citations:

Zbl 0567.12017; Zbl 0565.12014

References:

[1]	C. ARF, Untersuchungen über quadratische formen in Körpern der characteristik 2 (teil I), J. reine angew. Math., 183 (1941), 148-167. · JFM 67.0055.02
[2]	E. R. BERLEKAMP, An analog to the discriminant over fields of characteristic two, J. Algebra, 38 (1976), 315-317. · Zbl 0327.12101
[3]	N. BOURBAKI, Algèbre, Ch. 4 à 7, Masson, Paris, 1981. · Zbl 0498.12001
[4]	N. BOURBAKI, Algèbre, Ch. IX, Hermann, Paris, 1959.
[5]	C. CHEVALLEY, The algebraic theory of spinors, Columbia University Press, New-York, 1954. · Zbl 0057.25901
[6]	M. KNESER, Bestimmung des zentrums der cliffordschen algebren einer quadratischen form über einem Körper der characteristik 2, J. reine angew. Math., 193 (1964), 123-125. · Zbl 0056.25302
[7]	T. Y. LAM, The algebraic theory of quadratic forms, Benjamin, Reading (Mass.), 1973. · Zbl 0259.10019
[8]	Ph. REVOY, Remarques sur la forme trace, Linear Mult. Algebra, 10 (1981), 223-233. · Zbl 0466.10010
[9]	C.-H. SAH, Symmetric bilinear forms and quadratic forms, J. Algebra, 20 (1972), 144-160. · Zbl 0226.15010
[10]	J.-P. SERRE, Corps locaux, 2e éd., Hermann, Paris, 1968.
[11]	J.-P. SERRE, Extensions icosaédriques, Séminaire de théorie des nombres, exposé 19 (7 p.), Bordeaux, 1979-1980. · Zbl 0474.12011
[12]	J.-P. SERRE, L’invariant De Witt de la forme tr(x2), Comm. Math. Helvet., à paraître. · Zbl 0565.12014
[13]	T. SASAKI, Y. KANADA, S. WATANABE, Calculations of discriminants of high degree equations, Tokyo J. Math., 4 (1981), 493-499. · Zbl 0477.65034
[14]	J. TITS, Formes quadratiques, groupes orthogonaux et algèbres de Clifford. Invent. Math., 5 (1968), 19-41. · Zbl 0155.05202
[15]	A. R. WADSWORTH, Discriminants in characteristic two, Linear Mult. Algebra, à paraître. · Zbl 0567.12017