ideal number

ideal number

English

Etymology

Apparently a calque of German ideale Zahl, a concept in number theory developed by German mathematician Ernst Kummer (1810—1893) and later incorporated by Richard Dedekind into the ring theory concept of ideal.

Noun

ideal number (plural ideal numbers)

1. (number theory) An algebraic integer that represents an ideal in the ring of integers of a number field.
   • 1861, H. J. Stephen Smith, Report on the Theory of Numbers—Part II, Report of the Thirtieth Meeting of the British Association for the Advancement of Science, June-July 1860, British Association for the Advancement of Science, page 133,

     This symbolic representation of ideal numbers is very convenient, and tends to abbreviate many demonstrations.

     Every ideal number is a divisor of an actual number, and, indeed, of an infinite number of actual numbers. Also, if the ideal number ${\\displaystyle \\textstyle \\phi (\\alpha )}$ be a divisor of the actual number ${\\displaystyle \\textstyle {\\text{F}}(\\alpha )}$, the quotient ${\\displaystyle \\textstyle \\phi _{1}(\\alpha )={\\text{F}}(\\alpha )\\div \\phi (\\alpha )}$ is always ideal; for if ${\\displaystyle \\textstyle \\phi _{1}(\\alpha )}$ were an actual number, ${\\displaystyle \\textstyle \\phi (\\alpha )}$, which is the quotient of ${\\displaystyle \\textstyle {\\text{F}}(\\alpha )}$ divided by ${\\displaystyle \\textstyle \\phi _{1}(\\alpha )}$, ought also to be an actual number.

   • 1992 [World Scientific], C. Y. Hsiung, Elementary Theory of Numbers, 1995, Allied Publishers, page 214,

     One of the most important developments of algebraic number theory is the creation of ideal numbers. All ideal numbers can be classified into ideal number classes. Both the study of ideal numbers and the computation of ideal number classes are important problems of algebraic number theory. The introduction of ideal numbers was meant originally for the number theory, but now ideal numbers have become an important tool in algebra.

   • 1995, L. V. Kuz'min, Ideal number, entry in M. Hazewinkel (editor) Encyclopaedia of Mathematics: Volume 3, Springer, page 125,

     The semi-group D is a free commutative semi-group with identity; its free generators are called prime ideal numbers. In modern terminology, ideal numbers are known as integral divisors of A.

Translations

algebraic number that represents an ideal in the ring of integers

German: ideale Zahl f

See also

• perfect number (an unrelated concept)

Further reading

• Ideal Number on Wolfram MathWorld
• Ideal number on Encyclopedia of Mathematics