# Algebraic Number/Examples/Root of (2 plus Root 3)

## Example of Algebraic Number

$\sqrt {2 + \sqrt 3}$ is an algebraic number.

## Proof

Let $x = \sqrt {2 + \sqrt 3}$.

We have that:

 $\ds x^2$ $=$ $\ds 2 + \sqrt 3$ $\ds \leadsto \ \$ $\ds \paren {x^2 - 2}^2$ $=$ $\ds 3$ $\ds \leadsto \ \$ $\ds x^4 - 4 x^2 + 4$ $=$ $\ds 3$ $\ds \leadsto \ \$ $\ds x^4 - 4 x^2 + 1$ $=$ $\ds 0$

Thus $\sqrt {2 + \sqrt 3}$ is a root of $x^4 - 4 x^2 + 1 = 0$.

Hence the result by definition of algebraic number.

$\blacksquare$