Basis of the Identities of the Matrix Algebra of Order Two over a Field of Characteristic p ≠ 2

In this paper we prove that the polynomial identities of the matrix algebra of order 2 over an infinite field of characteristic p ≠ 2 admit a finite basis. We exhibit a finite basis consisting of four identities, and in “almost” all cases for p we describe a minimal basis consisting of two identities. The only possibilities for p where we do not exhibit minimal bases of these identities are p = 3 and p = 5. We show that when p = 3 one needs at least three identities, and we conjecture a minimal basis in this case. In the course of the proof we construct an explicit basis of the vector space of the central commutator polynomials modulo the ideal of the identities of the matrix algebra of order two.