Invariant ideals of abelian group algebras under the torus action of a field, I

Let V=V1⊕V2 be a finite-dimensional vector space over an infinite locally-finite field F. Then V admits the torus action of G=F• by defining g(v1⊕v2)=v1g−1⊕v2g. If K is a field of characteristic different from that of F, then G acts on the group algebra K[V] and it is an interesting problem to determine all G-stable ideals of this algebra. In this paper, we consider the special case when V1 and V2 are both 1-dimensional and we show that there are just four G-stable proper ideals of K[V].