@Carl Love thx for the answer but i really dont see a way Primfield could help, it calculates a primitive element of a field extension which in my case should not exist, if i understand the command correctly.

And for a more specific example:

I would like to know the minimal polynomial of s+t+1/(st) over the field Q( (u^3v^3-4uv^2+v^3+1)/(v(u-1)(v-1)(uv-1)) , (u^3v^3-3u^2v^3+6uv^2-3uv+v^3-3v^2+1)/(v(u-1)(v-1)(uv-1))) where u=s^2 and v=t^2.

Or of other similar problems