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Algebraic Number

A term in mathematics referring to a number that is a solution of a polynomial equation with integer coefficients. It can be expressed as a root of a polynomial equation of the form:

a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀ = 0

Senses and Usages:

1. Mathematics: In mathematics, an algebraic number is a complex number that satisfies a polynomial equation with integer coefficients.
2. Number Theory: Algebraic numbers play a critical role in number theory as they are used to study properties of integers and their relationships with other numbers.
3. Abstract Algebra: Algebraic numbers are closely related to abstract algebra, particularly field theory and Galois theory, which explore the roots and solvability of polynomial equations.

Examples:

1. The number √2 is an algebraic number since it satisfies the equation x² - 2 = 0.
2. π (pi) is not an algebraic number as it cannot be a root of any polynomial equation with integer coefficients. It is a transcendental number.

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