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9 hours ago comment added Bill Dubuque i.e. in $\,\Bbb Z_{13}\!:\ x = 0\,$ is clearly not a root of $\:\!f(x) = x^{101}-5\,$ since $\,f(0)= -5\neq 0.\,$ This argument is (implicitly) used in many places in algebra - as I explain in my answer. $\ \ $
9 hours ago history answered Shaun CC BY-SA 4.0