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paripari-gpfinite-field

In pari-gp, how to find a primitive element of finite field?

Say, I want a finite field containing q^n elements for some prime q and positive n. How to get its primitive element?

Solution

  • You can get some primitive element with the following code:

    var = 'x; \\ sets a variable in the polynomial representation of finite field
f = ffgen(ffinit(q, n)); \\ GF(q^n) ~ GF(q)[x]/<f(x)>. Note `f` is just an irreducible
a = ffprimroot(f); \\ gets a root `a` of `f`
poly = minpoly(a, var); \\ finds a minimal polynomial for `a`
primitive_elt = ffgen(poly, var); \\ finds a root of the minimal polynomial

\\ assertion: check the order
fforder(primitive_elt) == q^n-1

    Note, that the finite field can have certain primitive elements. The code above finds random any. Please, note that the code above assumes n > 1. Otherwise, the function minpoly crashes (tested with PARI/GP 2.7.2 32-bit).