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.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Field Extensions: $\S 38$. Simple Algebraic Extensions: Example $76$