Stable motivic π1π1 of low-dimensional fields

Let k be a field with cohomological dimension less than 3; we call such fields low-dimensional. Examples include algebraically closed fields, finite fields and function fields thereof, local fields, and number fields with no real embeddings. We determine the 1-column of the motivic Adams–Novikov spectral sequence over k  . Combined with rational information we use this to compute π1Sπ1S, the first stable motivic homotopy group of the sphere spectrum over k  . Our main result affirms Morel's π1π1-conjecture in the case of low-dimensional fields. We also determine π1+nαSπ1+nαS for weights n∈Z∖{−2,−3,−4}n∈Z∖{−2,−3,−4}.