Hi,

> The problem is that in fact you are *not* considering two polynomials: 
>   [...]
>
> Note that by saying 
>   S(x) = ... 
> you define S as a symbolic function on a symbolic variable x, and if you 
> re-define x later, then the variable of S will still be symbolic, and 
> not belong to a polynomial ring. 
>

Ok, my bad. I am still fairly new to Sage (even if I tried to submit a 
super small patch) and I don't completely understand what's the difference 
here, but I hope I will find the time to study this stuff about the types 
at the end of the semester.
 

> Is working in such a strange ring with zero divisors really what you want? 
>
> If you instead define 
> [...]
>
> Is this perhaps what you wanted to do? 
>

I suppose it is. At least, now my Euclidean algorithm seems to work quite 
correctly.

Unfortunately I am an undergraduate student in Computer Science (not even 
math) so there are some concepts which are still obscure to me. :P For 
example, what is the meaning of these two lines?

P.<x> = f[]
x = P.gen()

But the most important thing, thank you very much for helping me in solving 
the issue! I think you saved me a lot of time!

Best regards.

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