Why mentioning monic is important for gcd and lcm?

Question

Let $F$ is a field and $F[x]$ be the polynomial ring over $F$. Now in the definition of the gcd or lcm of any two polynomials $g(x)$ and $f(x)$ it is mentioned that the gcd or lcm are monic polynomials . My question is why this "$monic$" is important. If we have two polynomials with real coefficients, say $\ \ $$5x^{2}$$\ \ $ and $\ $$\ $ $25x$$\ \ $ then the lcm would be$\ \ $ $25x^{2}$$\ \ $ and the gcd is $\ \ $$5x$$\ \ $ none of which is monic. Then? Or did I got the gcd, lcm wrong?

Answer 1

If $F$ is a field, then, in $F[x]$, if $f$ divides $g$, then so does $\alpha f$ for any non-zero $\alpha \in F$. Requiring the gcd to be monic makes it unique. In your example with $F = \mathbb{R}$, $6x$ or $72x$ would be just as good values for $\gcd(5x^2, 25x)$ if we didn't adopt this convention (not all authors do). If the ring of coefficients is not a field, then you can't adopt this convention (and in the absence of some other convention the gcd is not unique: it is only determined up to multiplication by a unit): in $\mathbb{Z}[x]$ your answer of $5x$ as a greatest common divisor of $5x^2$ and $25x$ is correct and there is one other correct answer, namely $-5x$. Similar remarks apply to the lcm.