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In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed.
In mathematics, the ideal class group (or class group) of an algebraic number field K is the quotient group J K /P K where J K is the group of fractional ideals of the ring of integers of K, and P K is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of K.
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[1] Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science, and the social sciences. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation.
In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K.Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime ideals in K.
The word "class" in the term "equivalence class" may generally be considered as a synonym of "set", although some equivalence classes are not sets but proper classes. For example, "being isomorphic " is an equivalence relation on groups , and the equivalence classes, called isomorphism classes , are not sets.
German mathematician Carl Friedrich Gauss said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theory also studies the natural, or whole, numbers. One of the central concepts in number theory is that of the prime number , and there are many questions about primes that appear simple but whose ...
As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the values 3 and 1 nor between 4 and 4, that is, 3 < 1 and 4 < 4 both evaluate to false.