A primitive polynomial is a polynomial that generates all elements of an extension field from a base field. Primitive polynomials are also irreducible
polynomials. For any prime or prime
and any positive integer , there exists a primitive polynomial of degree over GF(). There are
primitive polynomials over GF(), where is the totient function.
A polynomial of degree over the finite field GF(2)
(i.e., with coefficients either 0 or 1) is primitive if it has polynomial
has order 3 since
to equation (◇), the numbers of primitive polynomials over GF(2) are
giving 1, 1, 2, 2, 6, 6, 18, 16, 48, ... (OEIS A011260) for ,
2, .... The following table lists the primitive polynomials (mod 2) of orders 1 through
Amazingly, primitive polynomials over GF(2) define a recurrence relation which can be used to obtain a new pseudorandom bit from the preceding ones.
See alsoFinite Field
, Irreducible Polynomial
, Primitive Element
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on Wolfram|AlphaPrimitive Polynomial
Cite this as:
Weisstein, Eric W. "Primitive Polynomial."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrimitivePolynomial.html