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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

 P(x)=a_nx^n+...+a_2x^2+a_1x+a_0

is of order n, denoted degP(x)=n. 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 the multiplicative order of a modulus). In particular, the order of a polynomial P(x) with P(0)!=0 is the smallest integer e for which P(x) divides x^e+1 (Lidl and Niederreiter 1994). For example, in the finite field GF(2), the order of x^5+x^2+1 is 31, since

 (x^(31)+1)/(x^5+x^2+1)=1+x^2+x^4+x^5+x^6+x^8+x^9 
+x^(13)+x^(14)+x^(15)+x^(16)+x^(17)+x^(20)+x^(21)+x^(23)+x^(26) (mod 2).

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

If p is an irreducible polynomial of degree k, then its order has to divide the order of the multiplicative group in the corresponding field extension, i.e., m^k-1 for modulus m.


See also

Irreducible Polynomial, Multiplicative Order, Polynomial Degree, Primitive Polynomial

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References

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

Referenced on Wolfram|Alpha

Polynomial Order

Cite this as:

Weisstein, Eric W. "Polynomial Order." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolynomialOrder.html

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