The homotopy groups of the {eta}-periodic motivic sphere spectrum

We compute the homotopy groups of the {eta}-periodic motivic sphere spectrum over a finite-dimensional field k with characteristic not 2 and in which -1 a sum of four squares. We also study the general characteristic 0 case and show that the {eta}-periodic slice spectral sequence over Q determines the {eta}-periodic slice spectral sequence over all extensions of Q. This leads to a speculation on the role of a connective Witt-theoretic J-spectrum in {eta}-periodic motivic homotopy theory.