Re: Primitive polynomials over GF(2^m)

Incidentally, both MathWorld and Wikipedia have
incorrect/inconsistent definitions of primitive
polynomials over finite fields.
...
Wikipedia:
In field theory, a branch of mathematics, a primitive
polynomial is the minimal polynomial of a primitive
element of the extension field GF(p^m). In other words,
a polynomial F(X) with coefficients in GF(p) = Z/pZ is
a primitive polynomial if it has a root in GF(p^m) such
that the linear span of \{1, \alpha, \alpha^2, \alpha^3,
\dots \} over GF(p) is the entire field GF(p^m), and
moreover, F(X) is the smallest degree polynomial having
alpha as root.

The "in other words" part of the definition is wrong,
but the following discussion is correct.

I've hopefully cleaned that up a bit. The primitive
element page that it linked too also only included
the general field definition.

In field theory, a branch of mathematics, a primitive
polynomial is the minimal polynomial of a primitive
element of the extension field GF(p^m). In other words,
a polynomial F(X) with coefficients in GF(p) = Z/pZ is
a primitive polynomial if it has a root a in GF(p^m) such
that {0,1,a,a^2,a^3,...,a^(p^m - 2)} is the entire field
GF(p^m), and moreover, F(X) is the smallest degree
polynomial having a as root.
.

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