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Algebraic number

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An algebraic number is a complex number that is a possible root of a non-zero polynomial function with rational coefficients and natural exponents.

Definition: The set of all algebraic numbers
\mathbb A = \left\{ \alpha \in \mathbb C \mid \exists P \in \mathbb Q \left[ x \right] : P\left(\alpha\right)=0, P\left(x\right)\ne 0 \text { for some }x\right\}

where \mathbb Q \left[ x \right] is the set of all polynomial functions with rational coefficients.

For example, if we take \alpha = \sqrt 3, \beta = 2i, and \gamma = \sqrt{2}+\sqrt[3]{3}, we will find these numbers to be algebraic since:

\alpha^2 - 3 = 0;
\beta^2 + 4 = 0;
\gamma^{6}-6\gamma^{4}-6\gamma^{3}+12\gamma^{2}-36\gamma+1=0.

Also, any rational number q is also algebraic, since it is a root of x-q.

Any number that is not algebraic, that is, not a root of any non-trivial polynomial function with rational roots, is defined to be transcendental.

In this way, constructing such a polynomial with a specific root proves that the given root is an algebraic number.

Algebraic numbers are either rational or irrational numbers, purely real or imaginary, or a complex combination. Essentially, most all complex numbers conceivable through simple algebraic relationships are algebraic, while the transcendentals exist between them and algebraically unrelated to them.

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