In abstract algebra, a matrix field is a field with matrices as elements. In field theory we come across two type of fields: finite field and infinite field. There are several examples of matrix fields of finite and infinite order. In general, corresponding to each field of numbers there is a matrix field.
There is a finite matrix field of order p for each positive prime p. One can find several finite matrix field of order p for any given positive prime p. In general, corresponding to each finite field there is a matrix field. However any two finite fields of equal order are algebraically equivalent. The elements of a finite field can be represented by matrices. In this way one can construct a finite matrix field.
Examples
1. The set of all diagonal matrices of order n over the field of rational (real or complex) numbers is a matrix field of infinite order under addition and multiplication of matrices.
2. The set of all diagonal matrices of order two over the field of integers modulo p (a positive prime) forms a finite matrix field of order p under addition and multiplication of matrices modulo p.