Therefore, the modern theory of Kummer may be stated in terms of divisors.
Already in the 1950s the concept of an ideal was generalized to the more comprehensive concept of a divisor. Subsequently, the concept of an ideal number was replaced by that of an ideal, equivalent to it, which can be described in terms of the field $ k $ itself. The class number can be explicitly described in terms of other field constants (the regulator, the discriminant and the degree of the field). Thus, any ideal number can be regarded as the $ h $ -th root of some element of the original field $ k $. He obtained the following important result: The class number $ h $ of $ k $ is finite, and the classes form an Abelian group under multiplication. The number of classes thus obtained is called the ideal class number of $ k $. Two ideal numbers are said to belong to the same class if their quotient belongs to $ k $. Kummer also introduced the concept of the ideal class number of a field $ k $. (Note that ideal numbers are defined relative to the original field $ k $ for another field $ k ^ \prime $ one must construct an extension $ K ^ \prime / k ^ \prime $ (of possibly different degree over $ k ^ \prime $ ) in which all ideal numbers of $ k ^ \prime $ are contained.) Two numbers in a field that differ by a unit (so-called associated numbers) have one and the same ideal factors. By introducing ideal numbers the theorem on unique factorization in $ k $ holds. These numbers were called ideal by Kummer (since they do not lie in the original field $ k $ ).
The concept of an ideal number arises from the fact that if a field $ k $ does not contain prime numbers into which any algebraic integer in $ k $ can be split, then there is a field $ K / k $ of finite degree over $ k $ in which there does exist a collection of numbers that play the role of primes for $ k $. In order to overcome this difficulty Kummer introduced ideal numbers, thus altering the entire structure of algebraic number theory for the future. This was Kummer's first point of view, but Dirichlet pointed out to him that unique factorization need not hold. If there would be unique factorization into prime factors in $ \mathbf Q ( \zeta ) $, then it would have been sufficient to prove that not all prime factors at the left-hand side occur with an exponent that is a multiple of $ p $. Kummer expanded the left-hand side using $ p $ -th roots of unity, and hence the problem was reduced to one about the algebraic integers of $ \mathbf Q ( \zeta ) $, $ \zeta $ a primitive $ p $ -th root of unity. The set of algebraic integers $ O _ $ in non-zero integers for any prime number $ p > 2 $.
The branch of number theory with the basic aim of studying properties of algebraic integers in algebraic number fields $ K $ of finite degree over the field $ \mathbf Q $ of rational numbers (cf.
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