The Gcdex function is a placeholder for the extended Euclidean algorithm applied to a and b which are polynomials in x over a field.  Gcdex computes g, the greatest common divisor of a and b, which is a monic polynomial in x.  Additionally s and t are (if present) assigned polynomials in x such that  $as+bt=g$  with $\mathrm{degree}\left(s\&comma;x\right)<\mathrm{degree}\left(b\&comma;x\right)$ and $\mathrm{degree}\left(t\&comma;x\right)<\mathrm{degree}\left(a\&comma;x\right)$. Gcdex is used in conjunction with either mod or evala as described below, both of which define the coefficient domain.

The call Gcdex(a, b, x, 's', 't') mod p performs the computation modulo p a prime integer. The multivariate polynomials a and b must have rational coefficients or coefficients in a finite field specified by RootOfs.

The call evala(Gcdex(a, b, x, 's', 't')) does likewise. The multivariate polynomials a and b must have algebraic number (or function) coefficients specified by RootOfs.

$\mathrm{Gcdex}\left(x^2+x+1\&comma;x^2-x+1\&comma;x\&comma;'s'\&comma;'t'\right)\mathbf{mod}11$

$1$

$s,t$

$5x+6,6x+6$

$\mathrm{alias}\left(\mathrm{sqrt2}=\mathrm{RootOf}\left(x^2-2\right)\right)\&colon;$

$\mathrm{evala}\left(\mathrm{Gcdex}\left(x^2-2\&comma;x^2-\mathrm{sqrt2}x\&comma;x\&comma;'s'\&comma;'t'\right)\right)$

$-\mathrm{sqrt2}+x$

$s,t$

$\frac{\mathrm{sqrt2}}{2},-\frac{\mathrm{sqrt2}}{2}$