Inconsistent behavior of gcd over non-prime field

[user202729]

Currently, over Z/6Z[x], gcd(x, 2) returns 1, while gcd(2x+2, 2x+2) raises exception.

While the documentation states that the result is tried to made monic when possible, I think the former result doesn't exactly make sense, since the ideal (x, 2) in Z/6Z[x] is not equal to the ideal (1).

Besides, the two results are inconsistent with each other.

Example code:

#include <flint/flint.h>
#include <flint/nmod_poly.h>

int main()
{
    nmod_poly_t A, B, G;

    nmod_poly_init(A, 6);
    nmod_poly_init(B, 6);
    nmod_poly_init(G, 6);

	/*
    nmod_poly_set_coeff_ui(A, 1, 1); // A = x
    nmod_poly_set_coeff_ui(B, 0, 2); // B = 2
	*/
    nmod_poly_set_coeff_ui(A, 1, 2); // A = 2*x+2
    nmod_poly_set_coeff_ui(A, 0, 2);
    nmod_poly_set_coeff_ui(B, 1, 2); // B = 2*x+2
    nmod_poly_set_coeff_ui(B, 0, 2);

    nmod_poly_gcd(G, A, B);

    flint_printf("GCD(x, 2) in Z/6Z[x] = ");
    nmod_poly_print_pretty(G, "x");
    flint_printf("\n");

    nmod_poly_clear(A);
    nmod_poly_clear(B);
    nmod_poly_clear(G);

    return 0;
}

[user202729]

This is actually harder than it looked, since Euclidean algorithm does not always work over rings.

E.g. gcd(x^2+x, 2*x+1) = 1 since (2*x+1)*(-4*x+1) ≡ 1 (mod x^2+x)