Ideal number

An element of the semi-group $D$ of divisors (cf. Divisor) of the ring $A$ of integers of an algebraic number field. 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 $A$. They can be identified in a natural way with the ideals (cf. Ideal) of $A$.

Ideal numbers were introduced in connection with the absence of uniqueness of factorization into prime factors in the ring of integers of an algebraic number field. For every $a\in A$, the factorization of the corresponding divisor $\phi(a)$ into the product of prime ideal numbers can be looked at as a substitute for unique factorization into prime factors if factorization in $A$ is not unique.

For example, the ring $A$ of integers of the field $\mathbf Q(\sqrt{-5})$ consists of the numbers $a+b\sqrt{-5}$ with integers $a$ and $b$. In this ring, the number 6 has two different factorizations:

$$6=2\cdot3=(1-\sqrt{-5})(1+\sqrt{-5}),$$

where the numbers 2, 3, $1-\sqrt{-5}$, and $1+\sqrt{-5}$ are pairwise non-associated irreducible (prime) elements of $A$; thus factorization into irreducible factors in $A$ is not unique. However, in $D$ the elements $\phi(2)$, $\phi(3)$, $\phi(1+\sqrt{-5})$, and $\phi(1-\sqrt{-5})$ are not irreducible; in fact, $\phi(2)=\mathfrak p_1^2$, $\phi(3)=\mathfrak p_2\mathfrak p_3$, $\phi(1-\sqrt{-5})=\mathfrak p_1\mathfrak p_2$, $\phi(1+\sqrt{-5})=\mathfrak p_1\mathfrak p_3$, where $\mathfrak p_1$, $\mathfrak p_2$ and $\mathfrak p_3$ are prime ideal numbers in $D$. Thus, the two factorizations of 6 into irreducible factors in $A$ give rise to one and the same factorization $\phi(6)=\mathfrak p_1^2\mathfrak p_2\mathfrak p_3$ in $D$.

The concept of an ideal number was introduced by E. Kummer in connection with his investigation of the arithmetic of cyclotomic fields (see , [2]). Let $K=\mathbf Q(\zeta)$ be the $p$-th cyclotomic field for some prime number $p$ and let $A=\mathbf Z[\zeta]$ be the ring of integers of $K$. The ideal numbers for $A$ were defined to be the products of prime ideal numbers, and the latter as the "ideal" prime divisors of natural prime numbers. To construct all the prime ideal numbers contained in a given natural prime number $q$, Kummer's theorem (cf. Kummer theorem) was used. Kummer used the fact that $A$ has basis $1,\zeta,\dots,\zeta^{p-2}$ over $\mathbf Z$ to investigate the factorization of the $p$-th cyclotomic polynomial $F_p(X)$ in the ring $(\mathbf Z/q\mathbf Z)[X]$. The ideal numbers dividing $q$ are in one-to-one correspondence with the irreducible factors of $F_p(X)$ in $(\mathbf Z/q\mathbf Z)[X]$ (the case $p=q$ required a somewhat different approach). A special method was applied to determine the exponent with which a given prime ideal number occurs in a given $a\in A$. He developed a similar method for creating a theory of divisibility in fields of the form $\mathbf Q(\zeta,m^{1/p})$, where $m\in\mathbf Q(\zeta)$.

The extension of the theory of ideal numbers to the case of arbitrary algebraic fields is due mainly to L. Kronecker and R. Dedekind. A division of the theory of ideal numbers into the theory of divisors (the approach of Kronecker) and the theory of ideals begins to appear in their papers. Dedekind associated with every ideal number a unique ideal of the ring $A$, which was defined by him as the subset of $A$ consisting of 0 together with all $a$ that are divisible by this ideal number. If $a_1,\dots,a_n$ are generators for the ideal $I$, then the ideal number corresponding to $I$ is the greatest common divisor of the ideal numbers $\phi(a_1),\dots,\phi(a_n)$.

Later, the concept of an ideal was extended to the case of an arbitrary ring $A$; rings for which the concepts of an ideal and a divisor coincide are now called Dedekind rings (cf. Dedekind ring).

References

[1a]	E. Kummer, "Zur Theorie der complexen Zahlen" J. Reine Angew. Math. , 35 (1847) pp. 319–326
[1b]	E. Kummer, "Ueber die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren" J. Reine Angew. Math. , 35 (1847) pp. 327–367
[2]	E. Kummer, "Mémoire sur la théorie des nombres complexes composés de racines de l'unité et de nombres entiers" J. Math. Pures Appl. , 16 (1851) pp. 377–498
[3]	H.M. Edwards, "The background of Kummer's proof of Fermat's last theorem for regular primes" Arch. Hist. Exact Sci. : 3 (1975) pp. 219–236
[4]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
[5]	N. Bourbaki, "Outline of the history of mathematics" , Springer (to appear) (Translated from French)