Central polynomials in the matrix algebra of order two

We exhibit minimal bases of the polynomial identities for the matrix algebra M2(K) of order two over an infinite field K of characteristic p≠2. We show that when p=3 the T-ideal of this algebra is generated by three independent identities, and when p>3 one needs only two identities: the standard identity of degree four and the Hall identity. Note that the same holds when the base field is of characteristic 0. Furthermore, using the exact form of the basis of the identities for M2(K) we give finite minimal set of generators of the T-space of the central polynomials for the algebra M2(K). The set of generators depends on the characteristic of the field as well.