Randpoly(n, x) mod p returns a polynomial of degree n in the variable x whose coefficients are selected at random from the integers mod p.

Randprime(n, x) mod p returns a random monic irreducible polynomial of degree $0<n$ in the variable x over the integers mod p where p must be a prime integer.

The optional third argument alpha specifies a representation for the finite field $\mathrm{GF}\left(p^k\right)$.  The field extension alpha is specified by a RootOf a monic univariate polynomial of degree k which must be irreducible.

Thus Randprime(n, x, alpha) mod p creates a random monic irreducible polynomial of degree $0<n$ in the variable x over $\mathrm{GF}\left(p^k\right)$.

$\mathrm{Randpoly}\left(4\&comma;x\right)\mathbf{mod}2$

$x^4+x$

$\mathrm{Randprime}\left(4\&comma;x\right)\mathbf{mod}2$

$x^4+x+1$

$\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left(y^2+y+1\right)\right)\&colon;$

$f≔\mathrm{Randpoly}\left(2\&comma;x\&comma;\mathrm{\alpha }\right)\mathbf{mod}2$

$f≔x^2+\mathrm{\alpha }+x$

$\mathrm{Factor}\left(f\right)\mathbf{mod}2$

$x^2+\mathrm{\alpha }+x$

$g≔\mathrm{Randprime}\left(2\&comma;x\&comma;\mathrm{\alpha }\right)\mathbf{mod}2$

$g≔\mathrm{\alpha }x+x^2+1$

$\mathrm{Irreduc}\left(g\right)\mathbf{mod}2$

$\mathrm{true}$