All one polynomial
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An AOP of degree m has all terms from xm to x0 with coefficients of 1, and can be written as
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 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 a primitive polynomial is x2 + x + 1.
Over , the AOP is irreducible whenever m + 1 is prime p, and therefore in these cases, the pth cyclotomic polynomial.
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