I am recently writing a math module about polynomials in python. And I encountered this question when it comes to compute the least common multiple of several polynomials.
Let [] denote lcm and let () denote gcd. Proof:
$$[f_1, f_2, f_3]=C*\frac{f_1f_2f_3(f_1, f_2, f_3)}{(f_1, f_2)(f_1, f_3)(f_2, f_3)}$$
Where C is a constant. I find it very similar to Inclusion–exclusion principle. And I can understand it by drawing Venn diagrams. But how to proof it rigorously?
Also, does the field of the coefficients matter? Will it be different in $\mathbb{Q}$, $\mathbb{R}$ or $\mathbb{C}$ ?
2026-04-09 07:43:53.1775720633
Find the least common multiple of three polynomials
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