A Note on Herm1t1an and Quadratic Forms

Let K be a field (characteristic # 2), and let L be a quadratic extension of K. Then L = K{yja) for some aeK. Let D be the quaternion algebra (a, b/K) generated by elements i, j satisfying i = a, j 2 = b, ij = —ji. We will assume that D is a division algebra, i.e. that the quadratic form <1> ~> ~b, ab} is anisotropic. Let — denote the standard involution on L and on D so that y/a = —*Ja on L and I = —i,j= —j on D. We will consider hermitian forms 0 over L and D with respect to the standard involution. Given such a hermitian form there is an underlying symmetric bilinear form over K given by •£($ + $). (Equivalently we have an underlying quadratic form over K given by taking (j)(x} x) for all x). Jacobson [1] proved that two hermitian forms over L, or over D, are isometric if and only if their underlying quadratic forms are isometric. We may ask when is a quadratic form over K the underlying form of some hermitian form over L or over D. This question is answered in the case of L by Milnor-Husemoller [2, Appendix 2], where they construct an exact sequence of W (K)-modu\QS