Summary: A HERMITIAN ANALOGUE OF THE BR ¨OCKER­PRESTEL THEOREM
VINCENT ASTIER AND THOMAS UNGER
ABSTRACT. The Br¨ocker­Prestel local-global principle characterizes weak iso-
tropy of quadratic forms over a formally real field in terms of weak isotropy
over the henselizations and isotropy over the real closures of that field. A
hermitian analogue of this principle is presented for algebras of index at most
two. An improved result is also presented for algebras with a decomposable
involution, algebras of pythagorean index at most two, and algebras over SAP
and ED fields.
1. INTRODUCTION
In the algebraic theory of quadratic forms over fields the problem of deter-
mining whether a form is isotropic (i.e., has a non-trivial zero) has led to the
development of several powerful local-global principles. They allow one to
test the isotropy of a form over the original field ("global" situation) by test-
ing it over a collection of other fields where the original problem is potentially
easier to solve ("local" situation).
The most celebrated local-global principle is of course the Hasse­Minkowski
theorem which gives a test for isotropy over the rational numbers Q in terms
of isotropy over the p-adic numbers Qp for each prime p and the real numbers
R. More generally, Q may be replaced by any global field F and the collection

Source: Astier, Vincent - Department of Mathematics, University College Dublin

Collections: Mathematics