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I want to prove that f(x) has a repeated root if and only if $<f(x)>+<f'(x)> \neq <1>$. I managed to prove that a repeated root implies $<f(x)>+<f'(x)> \neq <1>$, and I know how to prove that if $\gcd(f(x),f'(x))\neq 1$ then f(x) has repeated root. So it is left to show that $<f(x)>+<f'(x)> \neq <1>$ implies $\gcd(f(x),f'(x))\neq 1$, but I have no clue how to prove it.

I'll appreciate any lead.

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1 Answer 1

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Okay, the gcd of two polynomials is (by definition) a monic polynomial. So if $\gcd(f,f')\ne 1$, it must be a monic polynomial of degree $\geq 1$. This polynomial is both, a divisor of $f$ and a divisor of $f'$.

Conversely, if $\langle f\rangle +\langle f'\rangle \ne K[x]$, then $\langle f\rangle +\langle f'\rangle$ is a proper ideal of $K[x]$ and so an ideal of the form $\langle g\rangle$, since $K[x]$ is a principle ideal domain. Thus $af + bf'=g$ for some $a,b$. Hence, $g$ is a common divisor.

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