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6.3. Complex multiplication 173If the order O is non-maximal, then we can also associate to s the fractional ideal [s] O such that([s] O ) p = s p O p . Then [s] O is locally principal, and so is an invertible ideal of O. If s ≡ 1 mod fwhere f is the conductor of O, then we have [s] O = [s] OK ∩ O.6.3 Complex multiplication6.3.1 CM fieldDefinition 6.3.1 (CM field). A CM field is a totally imaginary extension of a totally real numberfield.Definition 6.3.2 (CM algebra). A CM algebra is a finite product of CM fields.An order in a CM algebra is nothing but a product of orders in the associated CM fields.When we speak of ideals in such an order, we mean lattices-ideals, i.e. ideals which are alsolattices, unless explicitly stated otherwise.If E is a CM algebra, then complex conjugation induces a positive involution on E whichdoes not depend on the complex embedding. We denote both the complex conjugation and thisinvolution by ·. Then, for any complex embedding φ, we have the identity φ = φ(·). Moreover, thecomplex embeddings of E come in pairs of complex conjugates whence the following definition.Definition 6.3.3 (CM type). Let E be a CM algebra. A CM type Φ on E is a subset ofHom(E, C) such thatHom(E, C) = Φ ⊔ Φ .We say that two CM types Φ and Φ ′ are equivalent if there exists an automorphism σ ∈ Aut(E)such thatΦσ = Φ ′ .If E ′ is a CM subalgebra of E and Φ is a CM type on E, then it induces a CM type Φ ′ on E ′ .We then say a CM type is primitive, or simple, if E is a CM field and there exists no such CMsubfield and induced type.Proposition 6.3.4 ([205, Proposition I.1.9], [159, Lemma 1.2.2]). For every CM pair (F, Φ)where F is a CM field, there exists a unique primitive pair (K, Ψ) which extends to (F, Φ).Moreover, K is the fixed field ofH = {σ ∈ Gal(L/Q) | Φ L σ = Φ L }where L is the Galois closure of F and Φ L is extended from Φ.A quartic CM field is either non-Galois, normal with cyclic Galois group — in which casesall CM types are primitive with respectively two and one equivalence classes of types — ornormal with Galois group isomorphic to C 2 × C 2 — in which case all types are non-primitive [252,Lemma I.3.4]. Hence we will speak of primitive, or non-primitive, quartic CM field.6.3.2 Reflex fieldDefinition 6.3.5 (Reflex field [205, Proposition I.1.16]). Let E be a CM algebra and Φ a CMtype. The reflex field E r of (E, Φ) is the fixed field ofH = { σ ∈ Gal(Q/Q) | σΦ = Φ } .