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144 Chapter 5. Complex multiplication and elliptic curvesDefinition 5.2.20 (Ideal quotient). Let K be an imaginary quadratic field and O an order inK. Let a and b be two fractional ideals. The ideal quotient, or ideal colon, of a and b, denoted by(a : b), is defined as(a : b) = {α ∈ K | αb ⊂ a} .It is also a fractional ideal.We remark that R(a) = (a : a).Definition 5.2.21 (Proper ideal). Let K be an imaginary quadratic field and O an order in K.A fractional ideal is said to be proper if its multiplier ring is exactly O. The set of proper idealsis denoted by Prop(O).The ring of integers O K is the maximal order of K, hence all its fractional ideals are proper.Definition 5.2.22 (Invertible ideal). Let K be an imaginary quadratic field and O an order inK. A fractional ideal a is said to be invertible if there exists a fractional ideal b such that ab = O.Moreover, let us define a −1 asa −1 = {α ∈ K | αa ⊂ O} = (O : a) .If a is invertible, then its inverse is a −1 . The set of invertible ideals is denoted by Inv(O).Equivalently, a fractional ideal is invertible if and only if it is projective (as a module).An invertible ideal a must verify R(a) = O, i.e. it must be proper, and its inverse 3 a −1 mustverify the same equality R(a −1 ) = O, i.e. it must also be proper. Furthermore, Frac(O) is asemigroup, Prop(O) is a semigroup and Inv(O) is a group.It is a specific feature to imaginary quadratic fields that proper ideals are always invertible.Proposition 5.2.23 ([160, Corollary of Theorem 8.1.2], [61, Proposition 7.4]). Let K be animaginary quadratic field and O an order in K. If a fractional ideal is proper, then it is invertible.Definition 5.2.24 (Principal ideal). Let K be an imaginary quadratic field and O an order inK. A fractional ideal is said to be principal if there exists α ∈ K ∗ such that a = αO. The set ofprincipal ideals is denoted by PF(O).The principal ideals are trivially invertible and so PF(O) is a subgroup of Inv(O).Definition 5.2.25 (Class groups). Let K be an imaginary quadratic field and O an order in K.The Picard group or class group of O is defined asPic(O) = Inv(O)/ PF(O) ,and its cardinality, the class number, by h(O).The proper class semigroup of O is defined asCl prop (O) = Prop(O)/ PF(O) ,and its cardinality, the proper class number, by H prop (O).The class semigroup of O is defined asCl(O) = Frac(O)/ PF(O) ,and its cardinality, the Kronecker class number, by H(O).3 We use the term inverse even though a could be non-invertible.