algebraic number

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Noun

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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 , whose coefficients are integers.
    The square root of a rational number, , is an algebraic number since it is a solution of the quadratic equation , whose coefficients are integers.
    • 1918, The American Mathematical Monthly, volume 25, Mathematical Association of America, page 435:
      Thus, the equation is satisfied for 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).

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