Field invariants

Many questions on sums of squares and quadratic forms over fields concern criteria for isotropy of quadratic forms. In this context one considers field invariants that are defined as suprema of the dimensions of anisotropic quadratic forms with a certain property. The u-invariant, the level and the pythagoras number are the most prominent examples of field invariants of this type. There are many open questions concerning the range of values for these invariants, the relations between them, and their behavior under field extension.

Invariants of real fields

We look in particular at real fields. There we study invariants related to the spaces of orderings (as the stability index) and various analogues of the u-invariant (generalized u-invariant of Elman-Lam, Hasse invariant, length).

• Becher, Leep. Real fields, valuations, and quadratic forms, Preprint 2011 (18p).
• Becher, Leep, Schubert. Semiorderings and stability index under field extensions, work in progress.
• Becher, Gładki. Symbol length and stability index, Journal of algebra to appear (6 p.).
• Becher, Leep. The Elman-Lam-Krüskemper Theorem, ISRN Algebra, Volume 2011 (2011), Article ID 106823 (8 pages).
• Becher, Leep. The length and other invariants of a real field, Mathematische Zeitschrift (to appear), Online First.
• Becher, Leep. Pythagoras numbers and quadratic field extensions, Conf. Proc. Algebraic and Arithmetic Theory of Quadratic Forms, Lago Llanquihue (Chile) 2007. Contemporary Math. 493 (2009).
• Becher. Totally positive extensions and weakly isotropic forms, Manuscripta Math. 120 (2006): 83-90.
• Becher. On fields of u-invariant 4, Archiv der Math. 86 (2006): 31-35.
• Becher. Minimal weakly isotropic forms, Mathematische Zeitschrift 252 (2006): 91-102.

Apart from these questions of a general nature, it is of course most desirable to compute the values of these invariants for familiar examples of fields, e.g. function fields over a finite field, the real or the rational numbers.

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