Polynomial Order

The highest order power in a univariate polynomial is known as its order (or, more properly, its polynomial degree). For example, the polynomial

is of order , denoted . The order of a polynomial is implemented in the Wolfram Language as Exponent[poly, x].

It is preferable to use the word "degree" for the highest exponent in a polynomial, since a completely different meaning is given to the word "order" in polynomials taken modulo some integer (where this meaning is the one used in themultiplicative order of a modulus). In particular, the order of a polynomial  with  is the smallest integer for which  divides  (Lidl and Niederreiter 1994). For example, in the finite field GF(2), the order of  is 31, since

This concept is closely related to that of the multiplicative order.

If  is an irreducible polynomial of degree , then its order has to divide the order of the multiplicative group in the corresponding field extension, i.e.,  for modulus .

REFERENCES:

Lidl, R. and Niederreiter, H. Introduction to Finite Fields and their Applications, 2nd ed. New York: Cambridge University Press, 1994.