All 2-dimensional algebras over any field have been classified. There are many references, beginning with the classification of Peirce. For a modern treatment see for example http://arxiv.org/pdf/1205.1221.pdf. There are two cases: characteristic two, and characteristic different from two. Implicitly such equivalence relations are treated, too.

The classification of associative algebras in low dimensions has a long history, beginning with work of Peirce. A good survey, and a classification of all nilpotent associative algebras of low dimension, over an arbitrary field, also of characteristic two, can be found in the paper of W. de Graaf, see http://arxiv.org/abs/1009.5339. I do not know if an "explicit list" of representatives is possible for the general case.

All 2-dimensional algebras over any field have been classified. There are many references, beginning with the classification of Pierce. For a modern treatment see for example http://arxiv.org/pdf/1205.1221.pdf. There are tow cases: characteristic two, and characteristic different from two. Implicitly such equivalence relations are treated, too.

All 2-dimensional algebras over any field have been classified. There are many references, beginning with the classification of Peirce. For a modern treatment see for example http://arxiv.org/pdf/1205.1221.pdf. There are two cases: characteristic two, and characteristic different from two. Implicitly such equivalence relations are treated, too.