$f,g,h$ are polynomials. Show that....

Question

Let $f,g$ and $h$ be polynomials. Show that $\gcd(f,g,h)=\gcd(\gcd(f,g),h)$.

I was thinking of signing $\gcd(f,g)=d$ and then write it by using Euclid's algorithm, but I couldn't get anything proper.

Answer 1

Can you show something along the following lines?

If $p$ is a polynomial that divides all of $f$, $g$ and $h$, will it also divide $\gcd(f,g)$ and $h$?

And vice versa, if $q$ is a polynomial that divides both $\gcd(f,g)$ and $h$, will it divide $f$, $g$, and $h$?

Answer 2

The proof of the associativity property of gcd follows immediately from the following

$\rm\qquad\quad\ \ \ d\mid (a,b,\ldots) \color{#c00}\iff d\mid a,b,\ldots\quad $ [Universal Property of GCD]

Hence $\rm\,\ \ d\mid((a,b),c)\color{#c00}\iff d\mid (a,b),\, c\color{#c00}\iff d\mid a,b,c $

Thus the associativity of GCD boils down to the associativity of AND (which is implicit in above, i.e. $\rm\ d\mid a,\,b,\ldots\,$ means $\rm\,d\mid a\,$ AND $\rm\,d\mid b,\ldots\,).$

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Compare analogous universal properties for more ell-known operations, e.g.

$\rm\qquad\quad\ \ \ S \supset A\cup B\color{#0a0}\iff S\supset A,B\quad $ [Universal Property of Union]

Hence $\rm\ \ \ S \supset (A\cup B)\cup C\color{#0a0}\iff S\supset A\cup B, C\color{#0a0}\iff S\supset A,B,C$