algebraic number

English

Noun

algebraic number (plural algebraic numbers)

(algebra, number theory) A complex number (more generally, an element of a number field) that is a root of a polynomial whose coefficients are integers; equivalently, a complex number (or element of a number field) that is a root of a monic polynomial whose coefficients are rational numbers.
The golden ratio (φ) is an algebraic number since it is a solution of the quadratic equation ${\\displaystyle x^{2}+x-1=0}$ , whose coefficients are integers.
The square root of a rational number, ${\\displaystyle \\textstyle {\\sqrt {\\frac {m}{n}}}}$ , is an algebraic number since it is a solution of the quadratic equation ${\\displaystyle nx^{2}-m=0}$ , whose coefficients are integers.
- 1918, The American Mathematical Monthly, Volume 25, Mathematical Association of America, page 435,
  Thus, the equation ${\\displaystyle x-e^{y}=0}$ is satisfied for ${\\displaystyle x=1,\\ y=0}$ and for no other pair of algebraic numbers.
- 1921, L. J. Mordell, Three Lectures on Fermat's Last Theorem, page 16,
  As a matter of fact, it is not true that the algebraic numbers above can be factored uniquely, but the first case of failure occurs when p = 23.
- 1991, P. M. Cohn, Algebraic Numbers and Algebraic Functions, Chapman & Hall, page 83,
  The existence of such 'transcendental' numbers is well known and it can be proved at three levels:
  (i) It is easily checked that the set of all algebraic numbers is countable, whereas the set of all complex numbers is uncountable (this non-constructive proof goes back to Cantor).