Algebraic Element of Ring Extension/Examples/Root 2

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Example of Algebraic Element of Ring Extension

$\sqrt 2$ is an algebraic element over the integers $\Z$.


We have that:

\(\ds x^2 - 2\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds \pm \sqrt 2\) Quadratic Formula

Thus $\sqrt 2$ is a root of $x^2 - 2$.

Hence the result by definition of algebraic element of $\Z$.