An ideal number is a formal factor introduced by Kummer to extend factorization in rings of cyclotomic integers
(Lemmermeyer 2009). The fundamental
theorem of arithmetic gives unique factorization
into primes for positive
integers, but factorization into irreducible
elements can fail to be unique in rings of algebraic
integers.
Dedekind generalized this approach using ideals (Lemmermeyer 2009). In modern terms, every nonzero proper
ideal in the ring of integers of a number
field factors uniquely, up to order, into nonzero prime ideals (Milne 2020, pp. 49 and 57).
See also
Divisor Theory,
Fermat's
Last Theorem,
Ideal
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References
Ferreirós, J. "Ideal Factors." §3.3.1 in Labyrinth
of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel,
Switzerland: Birkhäuser, pp. 95-97, 1999.Lemmermeyer, F. "Jacobi
and Kummer's Ideal Numbers." Abh. Math. Semin. Univ. Hambg. 79,
165-187, 2009. https://doi.org/10.1007/s12188-009-0020-5
and https://arxiv.org/abs/1108.6066.Milne,
J. S. Algebraic Number Theory. Version 3.08. Jul. 19, 2020. https://www.jmilne.org/math/CourseNotes/ANT.pdf.Referenced
on Wolfram|Alpha
Ideal Number
Cite this as:
Weisstein, Eric W. "Ideal Number." From
MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IdealNumber.html
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