We calculate the motivic stable homotopy groups of the two-complete sphere spectrum after inverting multiplication by the Hopf map over fields of cohomological dimension at most with characteristic different from (this includes the –adic fields and the finite fields of odd characteristic) and the field of rational numbers; the ring structure is also determined.
"The eta-inverted sphere over the rationals." Algebr. Geom. Topol. 18 (3) 1857 - 1881, 2018. https://doi.org/10.2140/agt.2018.18.1857