Nextpoly(f, x) mod p returns the next polynomial from f in x in lexicographical order over the integers modulo p. Similarly, Prevpoly(f, x) mod p returns the previous polynomial from f in x in lexicographical order over the integers modulo p.

Nextprime(f, x) mod p returns the next irreducible polynomial from f in x in lexicographical order over the integers modulo p. Similarly, Prevprime(f, x) mod p returns the previous irreducible polynomial from f in x in lexicographical order over the integers modulo p.

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, Nextprime(f, x, alpha) mod p computes the next irreducible polynomial from f in lexicographical order over $\mathrm{GF}\left(p^k\right)$.

$f≔x^4$

$f≔x^4$

$\mathrm{Nextpoly}\left(f\,x\right)\mathbf{mod}2$

$x^4+1$

$\mathrm{Nextprime}\left(f\,x\right)\mathbf{mod}2$

$x^4+x+1$

$\mathrm{Prevpoly}\left(f\,x\right)\mathbf{mod}2$

$x^3+x^2+x+1$

$\mathrm{Prevprime}\left(f\,x\right)\mathbf{mod}2$

$x^3+x^2+1$

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

$\mathrm{Nextpoly}\left(f\,x\,\mathrm{\alpha }\right)\mathbf{mod}2$

$x^4+1$

$\mathrm{Nextprime}\left(f\,x\,\mathrm{\alpha }\right)\mathbf{mod}2$

$x^4+\mathrm{\alpha }x+x^2+1$

$\mathrm{Prevpoly}\left(f\,x\,\mathrm{\alpha }\right)\mathbf{mod}2$

$\mathrm{\alpha }{x}^3+\mathrm{\alpha }{x}^2+x^3+\mathrm{\alpha }x+x^2+\mathrm{\alpha }+x+1$

$\mathrm{Prevprime}\left(f\,x\,\mathrm{\alpha }\right)\mathbf{mod}2$

$\mathrm{\alpha }{x}^3+\mathrm{\alpha }{x}^2+x^3+\mathrm{\alpha }x+x^2+\mathrm{\alpha }+x$