Algebraic Element of Ring Extension/Examples/Root 2

Example of Algebraic Element of Ring Extension

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

Proof

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 $\Z$.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 64$. Polynomial rings over an integral domain: Definition $(2)$