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 $aF+bG = d.$

How do I prove it? Please help me in this regard.

Thank you very much.

  • $\begingroup$ 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? $\endgroup$ – kesa Feb 12 at 11:07
  • $\begingroup$ Yeah you are correct @kesa. I have edited now accordingly. $\endgroup$ – Dbchatto67 Feb 12 at 14:15
  • $\begingroup$ What are your attempts to solve this problem? $\endgroup$ – kesa Feb 12 at 14:40

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