Editing All one polynomial (section)

==Properties==
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<ref>{{citation
 | last1 = Reyhani-Masoleh | first1 = Arash
 | last2 = Hasan | first2 = M. Anwar
 | contribution = On low complexity bit parallel polynomial basis multipliers
 | doi = 10.1007/978-3-540-45238-6_16
 | pages = 189–202
 | publisher = Springer
 | series = Lecture Notes in Computer Science
 | title = Cryptographic Hardware and Embedded Systems - CHES 2003
 | volume = 2779
 | year = 2003| doi-access = free
 | isbn = 978-3-540-40833-8
 }}.</ref>
*The AOP is [[irreducible polynomial|irreducible]] [[if and only if]] ''m'' + 1 is [[prime number|prime]] and 2 is a [[primitive root modulo n|primitive root]] modulo ''m'' + 1<ref name="hehcc"/> (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 (field theory)|primitive polynomial]] is ''x''<sup>2</sup> + 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]].<ref name="hehcc"/>

Over <math>\mathbb{Q}</math>, the AOP is irreducible whenever ''m'' + 1 is a prime ''p'', and therefore in these cases, the ''p''th [[cyclotomic polynomial]].<ref>{{citation
 | last1 = Sugimura | first1 = Tatsuo
 | last2 = Suetugu | first2 = Yasunori
 | doi = 10.1002/ecjc.4430740412
 | issue = 4
 | journal = Electronics and Communications in Japan
 | mr = 1136200
 | pages = 106–113
 | title = Considerations on irreducible cyclotomic polynomials
 | volume = 74
 | year = 1991}}.</ref>