Consider-a-polynomial-equation-i-0-n-a-i-x-i-0-a-i-Z-Prove-that-if-a-b-c-is-a-root-of-the-above-equation-then-a-b-c-is-also-a-root-a-b-c-Z-c-is-not-a-whole-square-

Question Number 3451 by prakash jain last updated on 13/Dec/15

$$\mathrm{Consider}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{equation} \\ $$$$\underset{{i}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{i}} {x}^{{i}} =\mathrm{0},\:{a}_{{i}} \in\mathbb{Z} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:{a}+{b}\sqrt{{c}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{root}\:\mathrm{of}\:\mathrm{the}\:\mathrm{above} \\ $$$$\mathrm{equation}\:\mathrm{then}\:{a}−{b}\sqrt{{c}}\:\mathrm{is}\:\mathrm{also}\:\mathrm{a}\:\mathrm{root}. \\ $$$${a},{b},{c}\in\mathbb{Z},\:{c}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{whole}\:\mathrm{square}. \\ $$

Commented by prakash jain last updated on 13/Dec/15

$${yes}. \\ $$

Commented by Filup last updated on 13/Dec/15

$${ah},\:\mathrm{i}\:\mathrm{see}.\:\mathrm{You}\:\mathrm{said}\:{c}\neq{whole}\:{number} \\ $$$$\mathrm{if}\:{c}\in\mathbb{Z},\:{c}\:{is}\:{integer}. \\ $$$$ \\ $$$$\mathrm{you}\:\mathrm{mean}\:\sqrt{{c}}\:\mathrm{is}\:\mathrm{not}\:\mathrm{whole}? \\ $$

Commented by Filup last updated on 13/Dec/15

$${a},\:{b}\in\mathbb{Z} \\ $$$${c}\in\mathbb{R} \\ $$

Commented by prakash jain last updated on 13/Dec/15

$${c}\in\mathbb{Z},\:\sqrt{{c}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{surd}\:\mathrm{or}\:\mathrm{complex}. \\ $$

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