Rost invariant on the center, revisited

TL;DR

This paper refines the understanding of the Rost invariant for algebraic groups by removing characteristic restrictions and simplifying formulas, while also extending classification results for invariants of quasi-trivial tori to all fields.

Contribution

It removes previous characteristic restrictions and ad hoc pairings in formulas for the Rost invariant, and extends the classification of invariants of quasi-trivial tori to all fields.

Findings

Formulas for the Rost invariant are now valid in all characteristics.

Simplified the formulas by removing ad hoc pairings.

Extended classification of invariants of quasi-trivial tori to all fields.

Abstract

The Rost invariant of the Galois cohomology of a simple simply connected algebraic group over a field $F$ is defined regardless of the characteristic of $F$, but unfortunately some formulas for it are only known with some hypothesis on the the characteristic. We improve those formulas by (1) removing the hypothesis on the characteristic and (2) removing an ad hoc pairing that appears in the original formulas. As a preliminary step of independent interest, we also extend the classification of invariants of quasi-trivial tori to all fields.