- All one polynomial
An

**all one polynomial**(AOP) is a polynomial used infinite field s, specifically GF(2) (binary). The AOP is a 1-equally spaced polynomial .An AOP of degree "m" has all terms from "x"

^{"m"}to "x"^{0}with coefficients of 1, and can be written as:$AOP(x)\; =\; sum\_\{i=0\}^\{m\}\; x^i$

or

:$AOP(x)\; =\; x^m\; +\; x^\{m-1\}\; +\; cdots\; +\; x\; +\; 1$

or

:$(x-1)AOP(x)\; =\; x^\{m+1\}\; -\; 1$

thus the roots of the

**all one polynomial**are allroots of unity .**Properties**Over GF(2) the AOP has many interesting properties, including:

*The

Hamming weight of the AOP is "m" + 1

*The AOP is irreducible if and only if "m" + 1 is prime and 2 is a primitive root modulo "m" + 1

*The only AOP that is aprimitive polynomial is "x"^{2}+ x + 1.Despite the fact that the Hamming weight is large, because of the ease of representation and other improvements there are efficient implementations in areas such as

coding theory andcryptography .Over $mathbb\{Q\}$, the AOP is irreducible whenever "m + 1" is prime p, and therefore in these cases, the "p"th

cyclotomic polynomial .**References**

*Wikimedia Foundation.
2010.*