Let $f(x) \in F[x]$ be irreducible (F is a Field). Suppose $f(x) \mid g_1(x)g_2(x)\dots g_k(x)$

Question

Let $f(x) \in F[x]$ be irreducible (F is a Field). Suppose $f(x) \mid g_1(x)g_2(x)\dots g_k(x)$

Prove $\exists i \in [1,k]$ where $f(x)|g_i(x)$

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(By contradiciton) Something along the lines

Assume $\forall i \in [1,k]$ where $f(x) \nmid g_i(x)$ (not div)

So, $$f(x) \nmid g_1(x),\dots, f(x) \nmid g_k(x)$$

$\therefore$ $f(x) \nmid g_1(x)*\dots*g_k(x) $ Contradiction!

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(Straight way)

Some thm

if $f(x)$ is irreducible and $f_1(x) \mid g_1(x)g_2(x)$ then $f(x)|g_1(x) $ or $f(x)|g_2(x)$

Using that thm $f(x)|g_1(x)$ or $f(x)|g_2(x)\dots g_k(x)$

$\vdots$

$f(x)\mid g_1(x)$or $f(x)\mid g_2(x) ...$ or $ f(x)\mid g_i(x) ...$ or $ f(x) \mid g_k(x)$

$\therefore$ $\exists i \in [1,k]$ where $f(x)|g_i(x)$

Something wrong specially with straight way??

Answer 1

Basis

if $f(x)$ is irreducible and $f_1(x) \mid g_1(x)g_2(x)$ then $$f(x)|g_1(x) \vee f(x)|g_2(x)$$ $\exists i \in [0,1] : f(x)|g_i(x)$

Induction For all Integers $k \geq 1$ assume $\exists i \in [0,k] : p(x)\mid g_i(x)$

in other words $f(x)\mid g_1(x) \vee \dots \vee f(x) \mid g_k(x)$

consider $f(x)\mid g_1*\dots*g_k(x)*g_{k+1}(x) $. So, $$ f(x)\mid g_1(x) \vee \dots \vee f(x)\mid g_{k}(x)g_{k+1}(x) $$ Applying Thm adds $f(x) | g_{k}(x) \vee f(x) | g_{k+1}(x) $

$\therefore$ $\exists i \in [0,k+1] : p(x)|g_i(x)$)