Research Projects

The proposed work is in the topic of central simple algebras over formally real fields. The behavior of the index of a central simple algebra over a field extension of its center is an interesting and difficult question. The case of central simple algebras of exponent 2 are particularly interesting because of their connections to quadratic forms theory. This is due to an important relation established by Merkurjev between the dimension of the anisotropic part of a quadratic form and the index of its Clifford invariant. In this project we propose to obtain properties of central simple algebras over formally real fields and pythagorean closure of a formally real field. K. J. Becher defined the notion of totally positive extensions, viz., a field extension L/K is said to be totally positive if a regular quadratic form q over K has a nontrivial zero over L then some n-orthogonal sum of q has a nontrivial zero over K. The pythagorean closure of a field is an example of a totally positive field extension. For a central simple algebra A over a formally real field F the pythagorean index of A is the index of A over the pythagorean closure of F. We propose to study the conjecture due to K. J. Becher which states that the pythagorean index of A is unchanged over a totally positive field extensions. Furthermore, we propose following two questions: Is every central simple algebra of exponent 2 over a formally real pythagorean field (i.e., every sum of a square is a square in a field) decomposable? If (A,σ) is a central simple algebra over a formally real pythagorean field with an orthogonal involution σ is weakly isotropic, i.e, if there are nonzero x_i in A such that ∑σ(x_i)x_i=0, then does it imply that σ(y)y=0 for some y in A? The last question is a generalization of a property quadratic forms over pythagorean fields.