All one polynomial

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

${\displaystyle AOP_{m}(x)=\sum _{i=0}^{m}x^{i}}$

or

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

or

${\displaystyle AOP_{m}(x)={x^{m+1}-1 \over {x-1}}.}$

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

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

• The Hamming weight of the AOP is m + 1, the maximum possible for its degree^[3]
• The AOP is irreducible if and only if m + 1 is prime and 2 is a primitive root modulo m + 1^[1] (over GF(p) with prime p, it is irreducible if and only if m + 1 is prime and p is a primitive root modulo m + 1)
• The only AOP that is a primitive polynomial is x² + 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 and cryptography.^[1]

Over ${\displaystyle \mathbb {Q} }$, the AOP is irreducible whenever m + 1 is a prime p, and therefore in these cases, the pth cyclotomic polynomial.^[4]