Cohomology of Lie algebras of polynomial vector fields on the line over fields of characteristic 2

For a field F, let Lk(F) be the Lie algebra of derivations f(t)ddt of the polynomial ring F[t], where f(t) is a polynomial of degree >k. For any k⩾−1, we present a basis of the space of the cohomology with finite-dimensional support of the Lie algebra Lk(F) with coefficients in the trivial module F for the case where char(F)=2. The main result obtained is an analog of the famous Goncharova's Theorem for the case char(F)=0 and k⩾1.