ideals of polynomial ring with complex number coefficients Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$.
Let $f,g\in \mathbb{C}[x,y]$. 
Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. 
Given $h\in \mathbb{C}[x,y]$, how to determine whether $h\in (f,g)$ or not? 
I have tried some examples by the online programming "sagemath".  
Are there any methods that can give a proof?
 A: You should use "Gröbner basis", (Groebner) . see the book by "Cox D., Little J., O'Shea D.": named "Ideals, Varieties, and Algorithms", for example. In page.82 they have:  
Corollary.2. Let $G = \{g_1, \cdots , g_t\}$ be a Groebner basis for an ideal $I \subset k[x_1, \cdots , x_n]$ and let $f \in k[x_1, \cdots , x_n]$. Then $f \in I$ if and only if the remainder on division of $f$ by $G$ is zero.  

 "Buchberger’s Algorithm", (page.88 of the book), helps you to produce the Groebner basis. Also "CoCoA" can compute Gröbner basis with the command "GBasis(I)" (for special field).

A: The algorithm is described in these notes by Madhu Sudan. The sage implementation is described in the Sage Manual.
A: Too long for a comment.
All CASes have bugs, so if you are using CAS solution, better
run on as many CASes as you can.
Comment suggests to use Groebner basis, but this leads to
the question "How do you compute Groebner basis without CAS?"
If your ideal is complicated enough, computing Groebner basis
by hand might not be trivial.
