The fundamental theorem of arithmetic states that every positive integer (except the number 1) can be represented in exactly
one way apart from rearrangement as a product of
one or more primes (Hardy and Wright 1979, pp. 2-3).
This theorem is also called the unique factorization theorem. The fundamental theorem of arithmetic is a corollary of the first of Euclid's
theorems (Hardy and Wright 1979).
For rings more general than the complex polynomials ,
there does not necessarily exist a unique factorization. However, a principal
ideal domain is a structure for which the proof of the unique factorization property
is sufficiently easy while being quite general and common.
Courant, R. and Robbins, H. What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, p. 23, 1996.Davenport, H. The
Higher Arithmetic: An Introduction to the Theory of Numbers, 6th ed. Cambridge,
England: Cambridge University Press, p. 20, 1992.Hardy, G. H.
and Wright, E. M. "Statement of the Fundamental Theorem of Arithmetic,"
"Proof of the Fundamental Theorem of Arithmetic," and "Another Proof
of the Fundamental Theorem of Arithmetic." §1.3, 2.10 and 2.11 in An
Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon
Press, pp. 3 and 21, 1979.Hasse, H. "Über eindeutige
Zerlegung in Primelemente oder in Primhauptideale in Integritätsbereichen."
J. reine angew. Math.159, 3-12, 1928.Lindemann, F. A.
"The Unique Factorization of a Positive Integer." Quart. J. Math.4,
319-320, 1933.Nagell, T. "The Fundamental Theorem." §4
in Introduction
to Number Theory. New York: Wiley, pp. 14-16, 1951.Zermelo,
E. "Elementare Betrachtungen zur Theorie der Primzahlen." Nachr. Gesellsch.
Wissensch. Göttingen1, 43-46, 1934.
![C[x]](/images/equations/FundamentalTheoremofArithmetic/Inline1.svg)
,
there does not necessarily exist a unique factorization. However, a principal
ideal domain is a structure for which the proof of the unique factorization property
is sufficiently easy while being quite general and common.
