The norm theorem for semisingular quadratic forms
Let \(\varphi\) be a quadratic form over a field \(F\), let \(X=(X_1,\ldots, X_n)\) be an \(n\)-tuple of variables and let \(p=p(X)\) be an irreducible polynomial in \(F[X]\). A classical problem in the algebraic theory of quadratic forms concerns finding necessary and sufficient conditions for \(p\) to be a similarity factor of \(\varphi\) over \(F(X)\), i.e., for \(\varphi\cong p\varphi\) to hold over \(F(X)\), in which case \(p\) is called a norm of \(\varphi\). Without loss of generality, one may assume \(\varphi\) to be anisotropic and \(p\) to have leading coefficient \(1\), say, with respect to the lexicographical ordering of monomials. If the characteristic of \(F\) is not \(2\), the answer was given by \textit{M. Knebusch} [Acta Arith. 24, 279--299 (1973; Zbl 0287.15010)]. He showed that \(p\) is a norm of \(\varphi\) if and only if \(\varphi\) becomes hyperbolic over \(F(p)\), the function field of \(p\), i.e., the field of fractions of the integral domain \(F[X]/(p)\). Actually, Knebusch proved an analogous result for bilinear forms in any characteristic (with hyperbolicity replaced by metabolicity), so it naturally carries over to quadratic forms in characteristic not \(2\) where bilinear and quadratic forms are essentially the same and metabolicity amounts to hyperbolicity. Let us from now on assume that the characteristic of \(F\) is \(2\) in which case the situation becomes more subtle. Here, a quadratic form \(\varphi\) decomposes as an orthogonal sum \(\varphi\cong R\perp\hbox{ql}(\varphi)\) where \(R\) is nonsingular and \(\hbox{ql}(\varphi)\) is the restriction of \(\varphi\) to its radical. In particular, \(R\) is isometric to an orthogonal sum of binary forms of type \([a,b]=ax^2+xy+by^2\), \(a,b\in F\) and thus \(\varphi\) is of dimension \(2r\) for some nonnegative integer \(r\), and \(\hbox{ql}(\varphi)\) is a diagonal form \(\langle a_1,\ldots,a_s\rangle=a_1x_1^2+\ldots+a_sx_s^2\), \(a_i\in F\), and is called the quasilinear part of \(\varphi\). The quasilinear part is uniquely determined by \(\varphi\), but \(R\) is in general not unique up to isometry if \(s>0\). So \(\varphi\) is nonsingular if \(s=0\), totally singular if \(r=0\), and it is called semisingular if \(r,s>0\). \textit{R. Baeza} [Commun. Algebra 18, No. 5, 1337--1348 (1990; Zbl 0711.11019)] showed that Knebusch's norm theorem carries over to characteristic not \(2\) for nonsingular \(\varphi\). If one considers semisingular or totally singular forms, one has to generalize the notion of hyperbolicity. The authors call a quadratic form \(\varphi\) (singular or not) quasi-hyperbolic if the dimension is even and if it contains a totally isotropic subspace of dimension \(\geq \frac{1}{2}\dim\varphi\). If \(\varphi\) is nonsingular, quasi-hyperbolicity coincides with the usual notion of hyperbolicity. Let now \(\varphi\) be totally singular and anisotropic over \(F\). Then the first author [Rocky Mt. J. Math. 36, No. 2, 575--592 (2006; Zbl 1142.11020)] has shown that \(p\) is a norm of \(\varphi\) iff \(\varphi\) becomes quasi-hyperbolic over \(F(p)\). (The reviewer [Contemp. Math. 344, 135--183 (2004; Zbl 1074.11023)] has independently shown a more general version of that result for diagonal forms of degree \(p\) in any positive characteristic \(p\).) In the remaining case of an anisotropic semisingular \(\varphi\), the first author and \textit{P. Mammone} [Indag. Math. New Ser. 17, No. 4, 599--610 (2006; Zbl 1130.11016)] have shown that \(p\) being a norm of \(\varphi\) implies the quasi-hyperbolicity of \(\varphi\) over \(F(p)\). The main result of the present paper establishes that the converse also holds, thus completing the picture concerning the norm theorem in characteristic \(2\). The main ingredients in the proof are a clever application of Scharlau's transfer maps in characteristic \(2\) in the one variable case and some specialization arguments to treat the multiple variable case.
function field of an irreducible polynomial
scientific article; zbMATH DE number 7317121
scientific article