prove: $\left(\frac{f(x)}{(f(x),g(x))},\frac{g(x)}{(f(x),g(x))}\right)=1$

Question

$f(x)$ and $g(x)$ are not all zeros. Prove:

$$\begin{align*}\left(\frac{f(x)}{(f(x),g(x))},\frac{g(x)}{(f(x),g(x))}\right)=1\end{align*}$$

prove:

we have $u(x)f(x)+v(x)g(x)=(f(x),g(x))$, and then divides $(f(x),g(x))$ in both sides, $u(x)\frac{f(x)}{(f(x),g(x))}+v(x)\frac{g(x)}{(f(x),g(x))}=1$，hence proved.

The question is that I'm not so sure, I can divide $(f(x),g(x))$ in both sides.

Dividing both sides by $(f(x),g(x))$ is fine, since you know that $(f(x),g(x))$ divides $f(x)$ and $g(x)$ by its definition. — user84413, Aug 17 '13 at 21:28

score 1 · Accepted Answer · answered Aug 17 '13 at 15:36

Things will be clearer if we start as follows:

Let the highest of powers of $h(x)$ that divides $f(x),g(x)$ be $a_f,a_g$ respectively.

WLOG , we can set $a_f\ge a_g\ge0$

Then the highest of powers of $h(x)$ that divides $(f(x),g(x))$ will be $a_g$ which clearly divides $f(x)$ and $g(x)$

This holds true for any divisor of $f(x)$ and/or $g(x)$

prove: $\left(\frac{f(x)}{(f(x),g(x))},\frac{g(x)}{(f(x),g(x))}\right)=1$

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