algebraic number

English

Noun

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 $x^{2}+x-1=0$, 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).

Hyponyms

• algebraic integer
  • phi, golden ratio

Coordinate terms

• transcendental number

Derived terms

• algebraic number theory

 

Translations

element of a number field that is a root of a polynomial with integer coefficients

• Chinese:

  Mandarin: 代數數／代数数 (zh) (dàishù shù)

• Czech: algebraické číslo n
• Finnish: algebrallinen luku
• Galician: número alxébrico (gl) m
• German: algebraische Zahl f
• Hungarian: algebrai szám (hu)
• Icelandic: algebruleg tala f, algebrutala f
• Italian: numero algebrico (it) m
• Kazakh: алгебралық сан (algebralyq san)
• Korean: 대수적 수(代數的數) (daesujeok su)
• Latin: numerus algebraicus m
• Romanian: număr algebric n
• Russian: алгебраическое число n (algebraičeskoje čislo)
• Serbo-Croatian: algebarski broj m
• Swedish: algebraiskt tal (sv) n
• Thai: จำนวนเชิงพีชคณิต (jam-nuuan-chəəng-pii-chá-ká-nít)
• Turkish: cebirsel sayılar pl

See also

• algebraic integer

Further reading

• 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