Algebraic number/Advanced

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An advanced level version of Algebraic number.

In number theory, an algebraic number is an element of a finite extension field of the rational numbers. Historically, such a field was often considered to be a subfield of the complex numbers, but several recent authors have dropped this requirement. An algebraic number must be a root of a polynomial with rational coefficients. Real or complex numbers that are not algebraic are called transcendental numbers.

Instances of algebraic numbers have been studied for for millennia as solutions of quadratic equations. They appear indirectly in the cakravāla method from the 11th century. In the 15th century, they arose in finding general solutions to cubic and quartic equations. However, the properties of algebraic numbers were not intensively studied until algebraic numbers appeared in an attempt to solve Fermat's last theorem.

The theory of algebraic numbers that ensued forms the foundation of modern algebraic number theory. Algebraic number theory is now an immense field, and one of current research, but so far has found few applications to the physical world.