The Classification of Three Dimensional Algebra

In this paper,3 dimensional algebra over a field K is discussed,and its structure and classification are offered.It is proved that: suppose A is a 3 dimensional algebra over a field K,then A is isomorphic to one of the following: the direct product K×K×K of three copies of K;the direct product of K and a 2 dimensional extension of K;a 3 dimensional extension of K;the direct product of K and the quotient algebra K[x]/(x~22) of the polynomial algebra K[x];the upper triangle 2×2 matrix algebra;the quotient algebra K[x]/(x~3) of the polynomial algebra;A has a square zero Jacobson radical of dimension 2.