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%I #9 Dec 18 2015 11:38:52
%S 1,2,4,6,8,9,10
%N Numbers n such that there is some field with u-invariant n.
%C Let F be a field of characteristic other than 2. Call a quadratic form over F isotropic if it represents zero nontrivially over F, or anisotropic otherwise. The u-invariant of a field F is the supremum of the dimensions of anisotropic quadratic forms over F.
%C It is folklore that 3, 5, and 7 are not in this sequence, see for example Proposition 6.8 in Lam 2005 chapter XI. Merkurjev showed that a(4) = 6, and more generally (see Merkurʹev 1991) that 2n is in this sequence for n > 0. Izhboldin showed that a(6) = 9. It is not known if 11, 13, 15, ... are in this sequence, see Question 6.4 in chapter XIII in Lam 2005.
%D Tsit-Yuen Lam, Fields of u-invariant 6 after A. Merkurjev, Ring theory 1989. In honor of S. A. Amitsur, Proc. Symp. and Workshop, Jerusalem/Isr. 1988/89, Isr. Math. Conf. Proc. 1, 12-30 (1989).
%D Tsit-Yuen Lam, Introduction to Quadratic Forms Over Fields (2005); 550 pp.
%D A. S. Merkurjev, Simple algebras over function fields of quadrics, manuscript (1989), 6 pp.
%D A. S. Merkurʹev, Simple algebras and quadratic forms (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 55:1 (1991), pp. 218-224; translation in Math. USSR-Izv. 38:1 (1992), pp. 215-221.
%H Oleg T. Izhboldin, <a href="http://www.jstor.org/stable/3062141">Fields of u-Invariant 9</a>, Annals of Mathematics, Second Series, 154:3 (Nov 2001), pp. 529-587.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Isotropic_quadratic_form">Isotropic quadratic form</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/U-invariant">u-invariant</a>
%e u(F) = 1 for all quadratically-closed fields (like C), so 1 is in the sequence.
%e u(F) = 2 for all finite fields (like F_2), so 2 is in the sequence.
%e u(F) is not 3 for any field F, so 3 is not in the sequence.
%K hard,nonn,more
%O 1,2
%A _Charles R Greathouse IV_, Oct 26 2015
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