u-invariant

In mathematics, the universal invariant or u-invariant of a field describes the structure of quadratic forms over the field.

The universal invariant u(F) of a field F is the largest dimension of an anisotropic quadratic space over F, or ∞ if this does not exist. Since formally real fields have anisotropic quadratic forms (sums of squares) in every dimension, the invariant is only of interest for other fields. An equivalent formulation is that u is the smallest number such that every form of dimension greater than u is isotropic, or that every form of dimension at least u is universal.

Examples

Properties

u(E) \le \frac{n+1}{2} u(F) \ .

In the case of quadratic extensions, the u-invariant is bounded by

u(F) - 2 \le u(E) \le \frac{3}{2} u(F) \

and all values in this range are achieved.[10]

The general u-invariant

Since the u-invariant is of little interest in the case of formally real fields, we define a general u-invariant to be the maximum dimension of an anisotropic form in the torsion subgroup of the Witt ring of F, or ∞ if this does exist.[11] For non-formally real fields, the Witt ring is torsion, so this agrees with the previous definition.[12] For a formally real field, the general u-invariant is either even or ∞.

Properties

References

  1. Lam (2005) p.376
  2. Lam (2005) p.406
  3. Lam (2005) p. 400
  4. Lam (2005) p. 401
  5. Lam (2005) p.484
  6. Lam, T.Y. (1989). "Fields of u-invariant 6 after A. Merkurjev". Ring theory 1989. In honor of S. A. Amitsur, Proc. Symp. and Workshop, Jerusalem 1988/89. Israel Math. Conf. Proc. 1. pp. 12–30. Zbl 0683.10018.
  7. Izhboldin, Oleg T. (2001). "Fields of u-Invariant 9". Annals of Mathematics. Second Series 154 (3): 529–587. Zbl 0998.11015.
  8. Lam (2005) p. 402
  9. Elman, Karpenko, Merkurjev (2008) p. 170
  10. Mináč, Ján; Wadsworth, Adrian R. (1995). "The u-invariant for algebraic extensions". In Rosenberg, Alex. K-theory and algebraic geometry: connections with quadratic forms and division algebras. Summer Research Institute on quadratic forms and division algebras, July 6-24, 1992, University of California, Santa Barbara, CA (USA). Proc. Symp. Pure Math. 58.2. Providence, RI: American Mathematical Society. pp. 333–358. Zbl 0824.11018.
  11. Lam (2005) p. 409
  12. 1 2 Lam (2005) p. 410
This article is issued from Wikipedia - version of the Monday, August 31, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.