Stable motivic pi_1 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 the first stable motivic homotopy group of the sphere spectrum over k. Our main result affirms Morels pi_1-conjecture in the case of low-dimensional fields. We also determine stable motivic pi_1 in integer weights other than -2, -3, and -4.