# All one polynomial

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An all one polynomial (AOP) is a polynomial used in finite fields, specifically GF(2) (binary). The AOP is a 1-equally spaced polynomial.

An AOP of degree m has all terms from xm to x0 with coefficients of 1, and can be written as

$AOP_m(x) = \sum_{i=0}^{m} x^i$

or

$AOP_m(x) = x^m + x^{m-1} + \cdots + x + 1$

or

$AOP_m(x) = {x^{m+1} - 1\over{x-1}}$

thus the roots of the all one polynomial of degree m are all (m+1)th roots of unity other than unity itself.

## Properties

Over GF(2) the AOP has many interesting properties, including:

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 and cryptography.

Over $\mathbb{Q}$, the AOP is irreducible whenever m + 1 is prime p, and therefore in these cases, the pth cyclotomic polynomial.