algebraic number

The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

English

English Wikipedia has an article on:

algebraic number

Wikipedia

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 $x^{2}+x-1=0$ , whose coefficients are integers.

The square root of a rational number, $\textstyle {\sqrt {\frac {m}{n}}}$ , is an algebraic number since it is a solution of the quadratic equation $nx^{2}-m=0$ , whose coefficients are integers.
- 1918, The American Mathematical Monthly, volume 25, Mathematical Association of America, page 435:
  Thus, the equation $x-e^{y}=0$ is satisfied for $x=1,\ y=0$ 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).