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Primes and Irreducibles in Truncation Integer Parts of Real Closed Fields

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Primes and Irreducibles in Truncation Integer Parts of Real Closed Fields Primes and Irreducibles in Truncation Integer Parts of Real Closed Fields. ¤ D. Biljakovic, M. Kochetov and S. Kuhlmanny 04. 06. 2004 Abstract Berarducci (2000) studied irreducible elements of the ring k((G<0))©Z, which is an integer part of the power series ¯eld k((G)) where G is an ordered divisible abelian group and k is an ordered ¯eld. Pitteloud (2001) proved that some of the irreducible elements constructed by Berarducci are actually prime. Both authors mainly con- centrated on the case of archimedean G. In this paper, we study truncation integer parts of any (non-archimedean) real closed ¯eld and generalize results of Berarducci and Pitteloud. To this end, we study the canonical integer part Neg (F ) © Z of any truncation closed sub¯eld F of k((G)), where Neg (F ) := F \ k((G<0)), and work out in detail how the general case can be reduced to the case of archimedean G. In particular, we prove that k((G<0)) © Z has (co¯nally many) prime elements for any ordered divisible abelian group G. Addressing a question in the paper of Berarducci, we show that every truncation integer part of a non-archimedean expo- nential ¯eld has a co¯nal set of irreducible elements. Finally, we apply our results to two important classes of exponential ¯elds: exponential algebraic power series and exponential-logarithmic power series. 0 Introduction An integer part (IP for short) Z of an ordered ¯eld K is a discretely ordered subring, with 1 as the least positive element, and such that for every x 2 K, there is a z 2 Z such that z · x < z + 1. It follows that the ring of integers Z is a convex subring of Z. If K is archimedean, then Z is the only IP of K, so we will be interested in the case of non-archimedean K. ¤1991 Mathematics Subject Classi¯cation: Primary 06F25; Secondary 13A16, 03H15, 03E10, 12J25, 13A05. ySecond author supported by an NSERC Postdoctoral Fellowship, third author partially sup- ported by an NSERC Discovery Grant. This paper was written while the third author was on sabbatical leave at Universit¶eParis 7. The authors wish to thank the Equipe de Logique de Paris 7 for its support and hospitality. 1 Shepherdson [S] showed that IP's of real closed ¯elds are precisely the models of a fragment of Peano Arithmetic called Open Induction (OI for short). OI is the ¯rst- order theory, in the language L := f+; ¢; <; 0; 1g, of discretely ordered commutative rings with 1 whose set of non-negative elements satis¯es, for each quanti¯er-free formula ©(x; y), the associated induction axiom I(©): 8y [©(0; y) and 8x [©(x; y) ! ©(x + 1; y)] ! 8x©(x; y)] : This correspondence led Shepherdson to investigate the arithmetic properties of IP's of real closed ¯elds. Given a ¯eld k and an ordered abelian group G, let us denote by k((G)) the ¯eld of generalized power series with exponents in G and coe±cients in k (see Section 1). We write k(G) := k(tg j g 2 G) for the sub¯eld generated by k and by the monomials tg. This sub¯eld is the quotient ¯eld of the group ring k[G] (which is a domain by the same argument as for the ring of polynomials). If k is an ordered ¯eld, then k((G)) can be ordered lexicographically. Shepherdson considered the countable recursive \algebraic Puiseux series ¯eld" Q(tg j g 2 Q)r, which can be viewed as a sub¯eld of the ¯eld Qr((Q)) of power series with coe±cients in the ¯eld of real algebraic numbers Qr and exponents in the additive group of rational numbers Q. (The superscript r denotes real closure.) He constructed an IP of this ¯eld in which the only irreducible elements are those in Z. In particular, the set of primes is not co¯nal in this model of OI. Thus the \in¯nity of primes" is not provable from Open Induction. On the other hand, subsequent to the work of Shepherdson, several authors (e.g. [M], [B{O], [Bi]) constructed various models of OI with (a co¯nal set of) in¯nite primes. In [M{R], Mourgues and Ressayre establish the existence of an IP for any real closed ¯eld K as follows. Let V be the natural valuation on K. Denote by k the residue ¯eld and by G the value group of K (see Section 1). [M{R] show that there is an order preserving embedding ' of K into the ¯eld of generalized power series k((G)) such that '(K) is a truncation closed sub¯eld (see Section 2 for details). They observe that for the ¯eld k((G)), an integer part is given by k((G<0)) © Z, where k((G<0)) is the (non-unital) ring of power series with negative support. It follows easily (see Proposition 2.1) that for any truncation closed sub¯eld F of k((G)), an <0 integer part is given by ZF = Neg (F ) © Z, where Neg (F ) := k((G )) \ F . We ¡1 shall call ZF the canonical integer part of F . Finally ' (ZF ) is an integer part of K if we take F = '(K). An integer part Z of K obtained in this way from a truncation closed embedding shall be called a truncation integer part of K. In [R], a proof for an exponential analogue of the main result of [M{R] is sketched: every exponential ¯eld (see Section 5) has an exponential integer part (EIP for short). An EIP is an IP that satis¯es some closure conditions under the exponential function (see Section 5). In [Bo2], Boughattas considers the following extension of OI in the language L[f2xg containing a symbol for the exponential function. He de¯nes OI(2x) to be the 2 ¯rst-order theory of discretely ordered commutative rings with 1 whose set of non- negative elements satis¯es, for each quanti¯er-free formula ©(x; y) of L [ f2xg, the associated induction axiom I(©), and also the following basic axioms for 2x: 20 = 1; 21 = 2; 2x+y = 2x2y; x < 2x: Analogously to Shepherdson's result, Boughattas constructs a model of OI(2x) in which the only irreducible elements are those in Z, and thus the set of primes is not co¯nal in this model of OI(2x). We note that, unlike the case of OI, an alge- braic description of models of OI(2x) is not known. In particular, the relationship between models of OI(2x) on the one hand and EIP's of exponential ¯elds on the other hand remains unclear. In [B; Concluding Remarks], Berarducci asks (and attributes the question to Ressayre) for an explicit axiomatisation of the class of EIP's of exponential ¯elds. We do not consider this question in this paper. Using a new kind of valuation whose values are ordinal numbers, Berarducci [B] studies irreducible elements in the ring k((G<0))©Z , that is, in the canonical IP of the power series ¯eld k((G)), focusing mainly on the case when G is archimedean. He gives a test for irreducibility based only on the order type of the support of a series. It is not known if every irreducible element of k((G)) is prime (that is, generates a prime ideal). Re¯ning the methods of [B], Pitteloud [P] shows that some of the irreducible series constructed in [B] are actually prime, in the case when G contains a maximal proper convex subgroup. In this paper, we extend the results of [B] and [P] to the canonical integer part Neg (F ) © Z of any truncation closed sub¯eld F , k(G) ½ F ½ k((G)) ; for an arbitrary divisible ordered abelian group G 6= 0. We shall denote by k[G<0] (respectively, by k[G·0]) the semigroup ring consisting of power series with neg- ative (respectively, non-positive) and ¯nite support. Note that k[G<0] ½ Neg (F ) since k(G) ½ F . [B; Theorem 11.2] says that, in the case of archimedean G, all irreducible elements of k[G<0]©Z remain irreducible in k((G<0))©Z. We study, for an arbitrary G, the behaviour of primes and irreducibles under the ring extensions k[G<0] © Z ½ Neg (F ) © Z ½ k((G<0)) © Z: (1) In [B; Concluding Remarks], the author asks whether every EIP of an exponential ¯eld contains a co¯nal set of irreducible elements. We give a partial answer to this question: applying our results to truncation EIP's of non-archimedean exponential ¯elds, we show that these EIP's indeed contain a co¯nal set of irreducible elements. Note that in this case, the rank of G (see Section 2) is a dense linearly ordered set without endpoints [K ; Corollary 1.23], in particular G cannot be archimedean, nor can it contain a maximal proper convex subgroup. 3 The structure of this paper is the following. In Section 1, we ¯x the notation and review the necessary background concerning ordered groups and ¯elds. In Sec- tion 2, we establish some straightforward facts used in the subsequent sections, in particular, to study the extensions (1). The main results of Section 3 are Proposi- tion 3.8 and Corollary 3.10 which provides (under mild conditions) co¯nal sets of irreducibles (=primes) in k[G<0] © Z. In Section 4, we generalize [B; Theorem 11.2] (see Corollaries 4.4 and 4.5) using the reduction to the case of an archimedean group G given in our Theorem 4.2 and Corollary 4.3 (some special cases of Corollary 4.3 already appeared in [B] and [P]).
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