Meaning of ideal number | Babel Free

Definitions

An algebraic integer that represents an ideal in the ring of integers of a number field.

Equivalents

Examples

“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⟳ ϕ(α) be a divisor of the actual number⟳ F(α), the quotient ϕ₁(α)=F(α)÷ϕ(α) is always ideal; for if ϕ₁(α) were an actual number⟳, ϕ(α), which is the quotient of F(α) divided by ϕ₁(α), 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.”

CEFR level