# Show that there exist $a,b \in K [X_1,X_2,\cdots,X_n]$ and $d \in K[X_1,X_2,\cdots,X_{n-1}]$ such that $aF+bG = d.$

Let $$K$$ be a field. Let $$F,G \in K [X_1,X_2,\cdots,X_n]$$ be two polynomials which are relatively prime to each other. Show that there exist polynomials $$a,b \in K [X_1,X_2,\cdots,X_n]$$ and $$0 \neq d \in K [X_1,X_2,\cdots,X_{n-1}]$$ such that $$aF+bG = d.$$

• Your question is odd: I can take $a, b, d$ to be $0$, then the equality hold trivially. It should rather be something like: for all $d$ there exist $a, b$.. Moreover: What is relatively prime in your definition? – kesa Feb 12 at 11:07