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Ordered Elds Ordered elds Francis J Rayner 16th January 2002 1. Introduction Let O denote the category of commutative totally ordered elds, and let R denote the category of commutative elds with a valuation in which the residue class eld is archimedean ordered, and where the morphisms preserve this order. There is a functor Λ from O to R(due to Krull () ) which is surjective on objects (Lang ( )). I describe below the set of all ordered elds F such that Λ(F ) = K for a xed eld K 2 obj(R). In particular, if K is maximally complete, the elds F form a single isomorphism class in O(Theorem 2 below). I do not know whether it is possible for there to be more than one isomorphism class in general. When K is maximally complete, it is isomorphic to a generalised eld of formal power series ( Kaplansky ( )) and then the isomorphism class of elds F contains a canonical generalised formal power series eld with real coecients and a natural (lexicographic) ordering. It follows that any ordered eld is order isomorphic to a subeld of a suitable formal power series eld over R, as stated by Gleyzal ( ) and by Fuchs ( ) but not proved by them. 2 Valuations and real places. Let F be a commutative eld, Γa totally ordered abelian group (the value group), and v : F ! Γ [ f1ga valuation (iea mapping satisfying v(xy) = v(x)v(y); v(x+y) ¸ maxfv(x); v(y)g and v(x) = 1 (, x = 0). Let B = fx : v(x) ¸ 0g;P = fx : v(x) > 0g:B is the valuation ring, P its maximal ideal. Let = B = P , the residue class eld. The natural map from B to extends to a map σ : F ! [ fg called the place associated with v(v is uniquely determined by σ). 3. From Orderings to places. Let F be a totally ordered commutative eld. Dene x; y 2 F to be archimedean equivalent if there exist m; n 2 N such that j x j < m j y j and j y j < n j x j. The element 0 2 F belongs to a single element equivalence class. Let Γ be the set of the remaining equivalence classes Γ; 1 the multiplication of F n f0g induces an abelian group structure on Γ(written additively), and the ordering of F induces a total ordering of Γ compatible with this group structure. Note that 0 2 Γ is the equivalence class containing the non-zero rational numbers, and that we have γ > 0 in Γ if each element of γ is innitesimal (i.e. x 2 γ , 8n 2 N n j x j < 1). The mapping v :F n f0g ! Γ is a valuation, and (with the notation of 2) by means of σ the ordering of F induces on an ordering, which is archimedean. (For, let x; y 2 B n P be such that σ(x); σ(y) are general non-zero elements of F. Then v(x) = 0 = v(y), so that x; y are in the same archimedean equivalence class of F , and the same is true of σ(x); σ(y) in F.) Thus Theorem 1 There is a functor Λ from O to R. This denition of Λ on objects was given by Krull ( ). Its extension to morphisms is immediate. 4.From real places to orderings. With the notation of 2, suppose that is totally ordered. The objective is to describe the total orderings of F compatible under σ with that of . Let ¤ s be the homomorphism s : F ! C2 for which s(x) = 1 for x > 0 and s(x) = 1 for x < 0: Let U = BP , so that U is the kernel of v restricted to F ¤. Note that σ maps U onto ¤, and write p = s ± σ. We have the commutative diagram 1p tM = 1C2 0 Γ0 with exact rows, where the existence of the group M and of the morphisms linking it to the rest of the diagram follow from the covariance of Ext(Γ; ):(See for instance, Maclane (15) p.66). The bottom row splits (Γ is torsion free, and C2 has exponent 2). Writing M = C2 © Γ,and t(x) = (t1(x); t2(x)), we have t2(x) = v(x) for all x 2 F , and t1(u) = p(u) for all u 2 U. Let S be the set of homomorphisms from F ¤ to M making the diagram commute, ¤ and let t 2 S. Then t(x) = q(x)t1(x) for some q 2 Hom(F ;C2), where q is trivial on U. Changing notation, we may regard q as lying in Hom(Γ;C2) and write t(x) = q(v(x))t1(x). Thus there is a bijection between Hom(Γ;C2) and S. ¤ For any f1 2 S dene x > 0 for x 2 F if f1(x) = 1; and x < 0 otherwise. To see that this gives a total order on F it is enough to prove that x > 0 implies 1 + x > O for any x 2 B. Either x 2 P or x 2 U. For x 2 P; σ(x) = 0; 1 + x 2 U; σ (1 + x) = 1; so that t1(1 + x) = s±σ(1 + x) = 1: For x 2 U, note that 1 + x 2 P is impossible for x > 0; so that σ(1 + x) 2 F, and then s ± σ(1 + x) = s(1 + σ(x)). Now f1(x) = 1 = s ± σ(x), so that 1 and σ(x) are both > 0 in the ordered eld . Hence 1 + σ(x) is also positive in so that t1(1 + x) = 1; as required. We have proved 2 Theorem 2:Hom(Γ;C2) acts freely and transitively on the set of distinct or- derings of a eld F compatible with a real place of F with associated value group Γ. Lang ((12), Theorem 1) showed that the set of distinct orderings of a eld with a real place is non-empty. Theorem 2 can be restated in terms of the Witt rings of F and . There is an epimorphism from W (F ) to W (), with kernel isomorphic to Hom(Γ;C2). The elements of W (K) are equivalence classes of quadratic forms, and are in bijection with the real closures of K, and so with the possible orderings of K. For details, see (13). 5. Maximally complete elds and Formal power series elds With the notation of 2, extension eld F1 of F is called an immediate extension of F if there exists a valuation of F1 which has the same value group and residue class eld as F , and whose restriction to F is the original valuation of F. A eld is called maximally complete if it admits no proper immediate extension. Hensel's lemma for maximally complete elds enables a polynomial equation over with a simple root to be lifted back across σ to a polynomial equation with a simple root in F. In particular a maxi- mally complete eld for which has characteristic 0 contains a coecient eld (a subeld F of F isomorphic to under the restriction to F of the place (F ! F¯ [ 1) associated with F.) (See Maclane (14)). Also for a maxi- mally complete eld, the group 1 + P is injective (closed under extraction of radicals). Kaplansky () showed that such a maximally complete eld is isomorphic to a eld of formal power series G(K; Γ; c), where c is the cocycle c(γ1; γ2) = σ ( t(γ1)t(γ2)=t(γ1 + γ2)) when t :Γ ! K is any map for which v ± t is the identity. The elements of G(K; Γ; c) are functions g from Γ to K with well ordered support fγ 2 Γ: g(γ) 6= 0g in Γ. With the P notation tγ for t(γ), it is convenient to write g as g(γ)tγ, and then the addition and multiplication (with the cocycle) are as this notation suggests. G(K; Γ; c) is a eld (the existence of multiplicative inverses was rst dealt with in (11) and later in a more general context in (6).) The valuation v(g) is the least element of the support of g, and the place σ(g) is g(0). This construction extends to ordered elds. Let F be an ordered eld for which a valuation and ordered residue class eld are constructed as in 3 above. The ordering of F has a unique prolongation to any given immediate extension F1 of F (For x 2 F1, there is y 2 F such that v(y) = v(x), and z2 F such that σ(z) = σ(xy¡1). Then xy¡1z¡1 2 1 + P , and the sign of x has to be the same as that of yz.) Extend the ordering to a maximal completion F1 of F. The Kaplansky construction of a formal power series eld applies, and if we choose the t(γ) to lie in F +(the strictly positive 3 elements of F ), the ordering in which a formal power series is positive if and only if its leading coecient is positive agrees with the ordering of F1. The role played by the cocycle can be claried from the following diagram. ! ! 1P ¡! ! ¡! ¡! ¡!¡! ¡!j+ ¡! where the existence of M and the morphisms linking it to the rest of the diagram follows from the covariance of Ext(Γ; ), as before. For a maximally complete eld, the groups 1 + P are injective, so that the columns split, and the middle row is determined by the last one.
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