algebraic number

algebraic number (plural algebraic numbers)

1. (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).

• algebraic integer
  • phi, golden ratio

• transcendental number

• algebraic integer

• Algebraic number field on Wikipedia.Wikipedia
• Algebraic integer on Wikipedia.Wikipedia
• Algebraic number on Encyclopedia of Mathematics
• algebraic number on nLab
• Algebraic Number on Wolfram MathWorld