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

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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.

$\blacksquare$


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