Matrix field

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.[1] 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.[2] 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.

See also

References

  1. ↑ Pandey, S. K. (2015). "Matrix Field of Finite and Infinite Order". International Research Journal of Pure Algebra 5 (12): 214–216.
  2. ↑ Lidl, Rudolf; Niederreiter, Harald (1986). Introduction to finite fields and their applications (1st ed.). Cambridge University Press. ISBN 0-521-30706-6.
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