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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.

little o
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  • 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 '19 at 11:07
  • Yeah you are correct @kesa. I have edited now accordingly. – little o Feb 12 '19 at 14:15
  • What are your attempts to solve this problem? – kesa Feb 12 '19 at 14:40

1 Answers1

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There may be a positive answer in here for $n = 3$. (I can't comment)

Trying to generalize it:

Since $k[x_1,...,x_n]$ is a UFD, $a,b$ as members of $k(x_1,...,x_{n-1})[x_n]$, a PID, have no common factors (Gauss's Lemma). Then you get $af + bg = 1$ and clearing denominators of $f,g$ you have the desired result.

Gustavo Labegalini
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