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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 $\mathbb{C}$ ?

Bill Dubuque
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JFR
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  • Venn diagram is a rigorous proof since the polynomial ring is UFD over those fields you consider. – Oolong Milktea May 06 '22 at 06:18
  • Depending on how you define the gcd/lcm for more than two elements, you need to prove that the operation is associative for the notation $(f_1,f_2,f_3)$ to even make sense (see this for example). Once you do that, the result follows from $[f_a,f_b],(f_a,f_b) = f_a f_b$ the same way you would prove the inclusion-exclusion principle in general. – dxiv May 06 '22 at 06:57

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