Ideal Classes of a Field

Every integer of a number field k determines a principal ideal; every fraction, i.e. every number к of k which is not an integer, can be represented as the quotient of two integers α and β and hence as the quotient of two ideals a and b: к = α/β = a/b. If we require that the ideals a and b have no common ideal factor then the representation of the fraction к as a quotient of ideals is uniquely determined. Conversely, if the quotient a/b of two ideals a and b — with or without a common factor — is equal to an integer or a fraction к= α/β of the field, we say that the two ideals a and b are equivalent to one another, denoted by a ~ b. If a/b = α/β then we have (β)a = (α)b and so we recognise that two ideals are equivalent to one another if and only if they are transformed into the same ideal when they are multiplied by suitable principal ideals. The totality of all ideals which are equivalent to a given ideal is called an ideal class. All principal ideals are equivalent to the principal ideal (1); the class which they form is called the principal class and is denoted by 1. If a ~ a′ and b′~ b′ then we have ab ~ a′b′. If A is the ideal class containing the ideal a and B is the ideal class containing b then the ideal class containing ab is called the product of the ideal classes A and B and is denoted by AB. Obviously 1B = B and conversely from AB = B it follows that A = 1.