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5.1. Further background on elliptic curves 133following fundamental result.Theorem 5.1.1 ([244, Theorem II.2.3], [128, Proposition II.6.8]). Let K be a perfect field. Letφ : C 1 → C 2 be a morphism between algebraic curves defined over K. Then φ is constant orsurjective (over K).Basic properties of morphisms between algebraic curves are defined using the canonicallyassociated maps on the corresponding function fields.Definition 5.1.2 (Function field). Let K be a perfect field and C algebraic curve defined on K.We denote by K(C) the function field of C.Definition 5.1.3. Let K be a perfect field. Let φ : C 1 → C 2 be a non-constant morphismbetween algebraic curves defined over K. Different homomorphisms between the function fields ofthe curves can be defined from a morphism between the curves:• φ ∗ : K(C 2 ) → K(C 1 ), g ↦→ g ◦ φ,• φ ∗ = (φ ∗ ) −1 ◦ N K(C1)/φ ∗ K(C 2).First, these maps can be used to completely characterize a morphism.Theorem 5.1.4 ([244, Theorem II.2.4], [128, Corollary I.6.12, Proposition II.6.8]). Let K be aperfect field. Let C 1 and C 2 be two curves defined over K.1. Let φ : C 1 → C 2 be a non-constant morphism. Then K(C 1 ) is a finite extension ofφ ∗ K(C 2 ).2. Let i : K(C 2 ) → K(C 1 ) be an injection leaving K fixed. Then there exists a uniquenon-constant morphism φ : C 1 → C 2 such that i = φ ∗ .3. Let K ′ ⊂ K(C 1 ) be a subfield of finite index containing K. Then there exists a smooth curveC ′ , unique up to K-isomorphism, and a morphism φ : C 1 → C ′ such that K ′ = φ ∗ K(C ′ ).Second, two very important quantities are defined using the function fields.Definition 5.1.5 (Degree). Let K be a perfect field. Let φ : C 1 → C 2 be a morphism betweentwo algebraic curves defined over K.If φ is constant, thenwe define its degree deg(φ) = 0.Otherwise, φ is surjective and its degree is defined asdeg(φ) = [K(C 1 ) : φ ∗ K(C 2 )],i.e. the dimension of the φ ∗ K(C 2 )-vector space K(C 1 ).The morphism φ is said to be separable, inseparable or purely inseparable, if the extension offunction fields is; the separable and inseparable degrees deg s (φ) and deg i (φ) are defined accordingly.Definition 5.1.6 (Ramification index). Let K be a perfect field. Let φ : C 1 → C 2 be a nonconstantmorphism between smooth curves defined over K. Let P ∈ C 1 . The ramification indexof φ at P is defined ase φ (P ) = ord P (φ ∗ t φ(P ) )where t φ(P ) ∈ K(C 2 ) is a uniformizer at φ(P ) (i.e. ord φ(P ) (t φ(P ) ) = 1).An application of these quantities is the study of the number of preimages of a point. Forexample, the degree is equal to the sum of ramification indices at the preimages of any point, andthe separable degree is equal to the number of preimages of a point except for a finite number ofthem [244, Proposition 2.6].