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class number 1. The class number of a number field is the cardinality of the ideal class group of the field. 2. In group theory, the class number is the number of conjugacy classes of a group. 3. Class number is the number of equivalence classes of binary quadratic forms of a given discriminant. 4. The class number problem. conductor
For instance, let be a root of + + =, then the ring of integers of the field () is [], which means all + with and integers form the ring of integers. An example of a nonprincipal ideal in this ring is the set of all + where and are integers; the cube of this ideal is principal, and in fact the class group is cyclic of order three.
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.
The integers arranged on a number line. An integer is the number zero , a positive natural number (1, 2, 3, . . .), or the negation of a positive natural number (−1, −2, −3, . . .). [1] The negations or additive inverses of the positive natural numbers are referred to as negative integers. [2]
Therefore, the ideal class group makes two fractional ideals equivalent if one is as close to being principal as the other is. The ideal class group is generally denoted Cl K, Cl O, or Pic O (with the last notation identifying it with the Picard group in algebraic geometry). The number of elements in the class group is called the class number of K.
In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite . [ 1 ] [ 2 ] The impolite numbers are exactly the powers of two , and the polite numbers are the natural numbers that are not powers of two.
The Montgomery form of the residue class a with respect to R is aR mod N, that is, it is the representative of the residue class aR. For example, suppose that N = 17 and that R = 100. The Montgomery forms of 3, 5, 7, and 15 are 300 mod 17 = 11, 500 mod 17 = 7, 700 mod 17 = 3, and 1500 mod 17 = 4.
Each residue class is an arithmetic progression, and thus clopen. Consider the multiples of each prime. These multiples are a residue class (so closed), and the union of these sets is all (Golomb: positive) integers except the units ±1. If there are finitely many primes, that union is a closed set, and so its complement ({±1}) is open.