The eta-inverted sphere over the rationals

Abstract

We calculate the motivic stable homotopy groups of the two-complete sphere spectrum after inverting multiplication by the Hopf map $\eta$ over fields of cohomological dimension at most $2$ with characteristic different from $2$ (this includes the $p$–adic fields ${\mathbb{Q}}_p$ and the finite fields ${\mathbb{F}}_q$ of odd characteristic) and the field of rational numbers; the ring structure is also determined.