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. © 2001 Academic Press.2411410434Asparouhov, T., Drensky, V., Koev, P., Tsiganchev, D., Generic 2 × 2 matrices in positive characteristic (2000) J. Algebra, 225, pp. 451-486De Concini, C., Procesi, C., A characteristic free approach to invariant theory (1976) Adv. in Math., 21, pp. 330-354Doubilet, P., Rota, G.-C., Stein, J., On the foundations of combinatorial theory (1974) Stud. Appl. Math., 3, pp. 185-216Drensky, V., A minimal basis of identities for a second-order matrix algebra over a field of characteristic 0 (1980) Algebra i Logika, 20, pp. 282-290Drensky, V., Computational techniques for PI-algebras (1990) Topics in Algebra Banach Center Publications, 26. , Warsaw: PWN. p. 17-44Drensky, V., Identities of representations of nilpotent Lie algebras (1997) Comm. Algebra, 25, pp. 2115-2127Filippov V., T., Varieties of Mal'tsev algebras (1981) Algebra i Logika, 20, pp. 300-314Formanek, E., The Polynomial Identities and Invariants of n × n Matrices (1991) CBMS Regional Conference Series Mathematics, 78. , Providence: American Mathematical SocietyGiambruno, A., Koshlukov, P., On the identities of the Grassmann algebras in characteristic p > 0 (2001) Israel J. Math.Koshlukov, P., Weak polynomial identities for the matrix algebra of order two (1997) J. Algebra, 188, pp. 610-625Koshlukov, P., Finitely based ideals of weak polynomial identities (1998) Comm. Algebra, 26, pp. 3335-3359Koshlukov, P., Ideals of identities of representations of nilpotent Lie algebras (2000) Comm. Algebra, 28, pp. 3095-3113Mal'tsev Yu., N., Kuz'min E., N., A basis for the identities of the algebra of second-order matrices over a finite field (1978) Algebra i Logika, 17, pp. 28-32Procesi, C., Computing with 2 × 2 matrices (1984) J. Algebra, 87, pp. 342-359Razmyslov Yu., P., Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero (1973) Algebra i Logika, 12, pp. 83-113Razmyslov Yu., P., Identities of Algebras and Their Representations (1994) Translations of Mathematical Monographs, 138. , Providence: American Mathematical SocietyRegev, A., On the Codimensions of Matrix Algebras (1988) Lecture Notes in Mathematics, 1352. , New York/Berlin: Springer-Verlag. p. 162-172Specht, W., Gesetze in Ringen, I (1950) Math. Z., 52, pp. 557-589Tki B., T., On the basis of the identities of the matrix algebra of second order over a field of characteristic zero (1981) Serdica, 7, pp. 187-194Vasilovsky S., Yu., The basis of identities of a three-dimensional simple Lie algebra over an infinite field (1989) Algebra i Logika, 28, pp. 534-554Vasilovsky, S., A finite basis for polynomial identities of the Jordan algebras of bilinear form (1991) Siberian Adv. Math., 1, pp. 1-43Vaughan-Lee M., R., Varieties of Lie algebras (1970) Quart. J. Math. Oxford (2), 21, pp. 297-308Vaughan-Lee M., R., Abelian-by-nilpotent varieties of Lie algebras (1975) J. London Math. Soc. (2), 11, pp. 263-266Zhevlakov K., A., Slin'ko A., M., Shestakov I., P., Shirshov A., I., Rings that Are Nearly Associative (1982) Pure and Applied Mathematics, 104. , New York: Academic Press