EMBEDDING ORDERED FIELDS IN

FORMAL POWER SERIES FIELDS

F.-V. Kuhlmann, S. Kuhlmann, M. Marshall, M. Zekavat

Let F be an ordered field, v the unique finest valuation on F compatible with the ordering (so v(a) ≥ v(b) iff n|b| ≥ |a| for some integer n ≥ 1), V the value group of v and κ the residue field of v (so κ is archimedean) [5] [10] [19]. If F′ is an ordered extension of F, then the finest valuation on F′ compatible with the extended ordering is an extension of v which we denote also by v. We denote by V′ and κ′ the value group and residue field of the extended v.

\overline{F} denotes the real closure of F. The residue field in this case is \overline{κ}, the real closure of κ. The value group is \overline{V} := the divisible hull of V. Any power series field is maximally complete [11] [21] [25]. \overline{κ}((\overline{V}))(\sqrt{−1}) = \overline{κ}(\sqrt{−1})((\overline{V})) is algebraically closed, so \overline{κ}((\overline{V})) is real closed. The natural valuation on \overline{κ}((\overline{V})), denoted also by v, is the unique finest valuation on \overline{κ}((\overline{V})) compatible with the ordering. The following is a summary of our results.

Suppose a proper embedding (see Section 1 for the definition) p : \overline{F} → \overline{κ}((\overline{V})) is given and F′ is an ordered extension of F generated by a single element y. We show there exists a canonically defined power series ϕ ∈ \overline{κ′}((\overline{V′})) such that p extends to a proper embedding p : \overline{F′} → \overline{κ′}((\overline{V′})) via y ↦ ϕ. Using results of Mourgues and Ressayre [16] we show that p(\overline{F}) truncation closed ⇒ p(\overline{F′}) truncation closed. We also show that p(F) ⊆ κ̃((V)) ⇒ ϕ ∈ κ̃′((V′)) (κ̃ denotes the smallest subextension κ ⊆ κ̃ ⊆ \overline{κ} closed under adjoining n-th roots of positive elements, n ≥ 1. Of course, if κ = ℝ, then κ = κ̃ = \overline{κ}). Roughly, this is what is asked for by MacLane and Schilling in [14, Final Remark]. It allows us to “read off” the extended value group V′ directly from the power series ϕ.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 12J10, 12J15, 14P99.

The research of each of the first three authors was supported in part by NSERC of Canada.

The research of the third author was also supported by a sabbatical stipend from Universidad Complutense in Madrid. The research of the fourth author was supported in part by a scholarship from the government of Iran.

In Section 5 we apply this result to show that, modulo natural constraints, the extended value group V′ can be prescribed arbitrarily.

In Sections 3 and 4 we point out a natural one-to-one correspondence between orderings on the polynomial ring \overline{κ}[Y_{1}, ..., Y_{ℓ}], κ a given archimedean ordered field, and certain distinguished ℓ-tuples (ϕ_{1}, ..., ϕ_{ℓ}), ϕ_{i} ∈ ℝ((ℍ_{i})), i = 1, ..., ℓ where ℍ_{0} ↪ ... ↪ ℍ_{ℓ}, ℍ_{i} is the product of 2^{i} − 1 copies of (ℝ, +, ≤) ordered lexicographically, and “read off” properties of the ordering from the associated ℓ-tuple. Orderings on \overline{κ}[Y_{1}, ..., Y_{ℓ}] also correspond to ultrafilters of semialgebraic sets in \overline{κ}^{ℓ} [4] [5] so another way of describing this result is to say that we have associated a distinguished “ideal point” (ϕ_{1}, ..., ϕ_{ℓ}) to each such ultrafilter.

In Section 6, we apply the analysis in Sections 1 and 2 to investigate which cuts are realized in a real closed field. In particular we give a valuation theoretic characterization of ℵ_{α}–saturated real closed fields, for a cardinal ℵ_{α} ≥ ℵ_{0} (cf. Theorem 6.2 below).

The results of Sections 1 and 2 are needed throughout the remainder of the paper, and Section 4 depends on Section 3 as well. Otherwise, the sections are independent of each other. After the first two sections, the reader could continue either with Sections 3 and 4, or with Section 5, or go directly to Section 6, depending on the interest.

The authors want to thank M.E. Alonso and C. Andradas, for sharing their expertise with us. The original motivation comes from the paper [3] by M.E. Alonso, J.M. Gamboa and J.M. Ruiz. The focus in [3] (also see [2]) is on describing orderings on ℝ(Y_{1}, ..., Y_{ℓ}) in terms of analytic germs ψ : (0, ϵ) → ℝ^{ℓ}.

1. The Embedding Theorem.

We use the terminology and notation introduced above. Any embedding p : \overline{F} → \overline{κ}((\overline{V})) is order preserving and consequently also valuation preserving. Since \overline{κ} is archimedean, the induced embedding of the residue fields \overline{κ} ↪ \overline{κ} is the identity mapping. We say that an embedding p : \overline{F} → \overline{κ}((\overline{V})) is proper if the induced embedding of value groups \overline{V} ↪ \overline{V} is the identity mapping, i.e., if v(p(z)) = v(z) for all z ∈ \overline{F}. The support of a power series a = ∑_{γ∈\overline{V}} a_{γ}x^{γ} ∈ \overline{κ}((\overline{V})) is the set \{γ ∈ \overline{V} | a_{γ} ≠ 0\}. This is well-ordered by definition. A power series b = ∑_{γ∈\overline{V}} b_{γ}x^{γ} is a truncation of a power series a = ∑_{γ∈\overline{V}} a_{γ}x^{γ} if there exists γ_{0} ∈ \overline{V} ∪ \{∞\} such that

b_{γ} = \{ \begin{matrix}a_{γ} & if γ < γ_{0} \\ 0 & if γ ≥ γ_{0} \end{matrix} .

We write b = (a)_{<γ_{0}} in this case. A subfield K of a power series field \overline{κ}((\overline{V})) is said to be truncation closed if every truncation of every element of K belongs to K [16].

1.1 Theorem. Suppose F′ = F (y), an ordered extension of F, and p : \overline{F} → \overline{κ}((\overline{V})) is a proper embedding. Then there is a canonically defined power series ϕ ∈ \overline{κ′}((\overline{V′})) (depending on y) such that:

(1) p extends to a proper embedding p : \overline{F′} → \overline{κ′}((\overline{V′})) via y ↦ ϕ.

(2) If p(\overline{F}) is truncation closed, then so is p(\overline{F′}).

(3) If p(F) ⊆ κ̃((V)), then ϕ ∈ κ̃′((V′)). Equivalently, p(F′) ⊆ κ̃′((V′)); equivalently, V′ = V_{1} and κ̃′ = κ̃_{1} where V_{1} is the group generated by V and the support of ϕ and κ_{1} is the field generated by κ and the coefficients of ϕ.

The definition of ϕ and the proof of part (1) of Theorem 1.1 (see below) seem to be part of the folklore. Also, as we will show, (2) is a consequence of the results in [16]. In this sense, the main new result here is (3). It does not seem possible to improve on (3) in any substantial way. For example, it is not true in general that p(F) ⊆ κ((V)) ⇒ ϕ ∈ κ′((V′)).

The equivalence of the various conclusions in (3) is pretty clear: Since p(F′) = p(F (y)) = p(F)(ϕ) and p(F) ⊆ κ̃((V)), it is clear that p(F′) ⊆ κ̃′((V′)) iff ϕ ∈ κ̃′((V′)). From the definition of V_{1} and κ_{1} it is clear that p(F′) ⊆ κ̃_{1}((V_{1})) so, comparing value groups and residue fields, V′ ⊆ V_{1} and κ′ ⊆ κ̃_{1} (so κ̃′ ⊆ κ̃_{1}). If p(F′) ⊆ κ̃′((V′)) then we obviously have the reverse inclusions, so V′ = V_{1} and κ̃′ = κ̃_{1}. Conversely, if V′ = V_{1} and κ̃′ = κ̃_{1} then p(F′) ⊆ κ̃_{1}((V_{1})) = κ̃′((V′)).

We have the following easy consequence of Theorem 1.1 (cf. also [20], Satz 27, p. 118, and Satz 34, p. 124):

1.2 Corollary. There exists a proper embedding p : \overline{F} → \overline{κ}((\overline{V})) such that (1) p(\overline{F}) is truncation closed.

(2) p(F) ⊆ κ̃((V)).

Proof. This is immediate from Theorem 1.1, using Zorn’s Lemma. □

1.3 Definition. The definition of ϕ is complicated by the fact that there are four cases to consider. If y is algebraic over F, there is a unique F -embedding i : F (y) → \overline{F} preserving the ordering. Define ϕ = p(i(y)) in this case. We refer to this as the algebraic case. Suppose now that y is transcendental over F. The ordering on F (y) extends uniquely to \overline{F (y)} and, consequently, to the intermediate field \overline{F} (y). Let

S = \{v(y − z) | z ∈ \overline{F} \}.

We define an element w in \overline{κ}((\overline{V})) as follows: Suppose γ ∈ \overline{V}. If γ ≥ δ for all δ ∈ S define w_{γ} = 0. If γ < δ = v(y − z) for some z ∈ \overline{F}, define w_{γ} to be the coefficient of x^{γ} in p(z).

(Observe: w_{γ} is independent of the choice of z. If we also have γ < δ′ = v(y − z′), z′ ∈ \overline{F}, then v(p(z) − p(z′)) = v(z − z′) ≥ min\{δ, δ′\} > γ so the coefficient of x^{γ} in p(z) and p(z′) are the same.) Define w = ∑_{γ∈\overline{V}} w_{γ}x^{γ}.

In what follows, p(z)_{γ} denotes the coefficient of x^{γ} in p(z). For z ∈ \overline{F}, p(z) ∈ \overline{κ}((\overline{V})) so \{γ ∈ \overline{V} | p(z)_{γ} ≠ 0\} is well-ordered. Consequently, \{γ ∈ \overline{V} | w_{γ} ≠ 0\} is also well-ordered so w is indeed an element of \overline{κ}((\overline{V})). Note that w_{γ} = p(z)_{γ} for all γ < v(y − z). Hence, v(w − p(z)) ≥ v(y − z).

Either S has no largest element or S has a largest element. If S has no largest element, we define ϕ = w. We refer to this as the immediate transcendental case. Claim 1. In this case, w ∉ p(\overline{F}). For, if z ∈ \overline{F}, there exists z′ ∈ \overline{F} such that v(y − z) < v(y − z′). Then v(p(z′) − p(z)) = v(z′ − z) = v(y − z) and v(w − p(z′)) ≥ v(y − z′) > v(y − z). Consequently, v(w − p(z)) = v(y − z) ≠ ∞ so w ≠ p(z).

This case does not occur if p(\overline{F}) = \overline{κ}((\overline{V})). Suppose now that S has a largest element γ. Thus there exists z ∈ \overline{F} such that v(y − z) = γ and w is a truncation of p(z). There are two subcases: Either γ ∈ \overline{V} or γ ∉ \overline{V}. Suppose γ ∉ \overline{V}. In this case we define ϕ = w + x^{γ} if y > z and ϕ = w − x^{γ} if y < z. Claim 2. This definition does not depend on the choice of z. If we also have z′ ∈ \overline{F}, v(y − z′) = γ then v(z − z′) ≥ γ. Since γ ∉ \overline{V}, this forces v(z − z′) > γ, so y − z′ and y − z have the same sign.

We refer to this as the value transcendental case. Finally, suppose γ ∈ \overline{V}. Claim 3. In this case, there is a unique element r ∈ ℝ\\overline{κ} such that if z ∈ \overline{F}, v(y − z) = γ, then z < y iff p(z) < w + rx^{γ} and z > y iff p(z) > w + rx^{γ}. To prove this, fix z_{0} ∈ \overline{F} such that v(y − z_{0}) = γ and a ∈ \overline{F} such that v(a) = γ, a > 0. Let r = p(z_{0})_{γ} + sp(a)_{γ} where s ∈ \overline{κ′} is the image of \frac{y−z_{0}}{a} in the residue field. Since γ is the maximal element of S, s ∉ \overline{κ}, so r ∉ \overline{κ}. (If s ∈ \overline{κ} then s is the image of some b ∈ \overline{F}, v(b) = 0. Then v(\frac{y−z_{0}}{a} − b) > 0 so v(y − (z_{0} + ab)) > γ, contradicting the definition of γ.) Suppose z ∈ \overline{F}, v(y − z) = γ. If z < y, then \frac{y−z}{a} = \frac{y−z_{0}}{a} + \frac{z_{0}−z}{a} > 0 so, going to the residue field, s + \frac{p(z_{0})_{γ}−p(z)_{γ}}{p(a)_{γ}} > 0, i.e., p(z)_{γ} < r, so p(z) < w + rx^{γ}. Similarly, if z > y, then p(z)_{γ} > r so p(z) > w + rx^{γ}. Finally, if we take z = z_{0} + ta, t ∈ ℚ, we see that as t runs through ℚ, p(z)_{γ} = p(z_{0})_{γ} + tp(a)_{γ} runs through a dense subset of \overline{κ}. This implies r is unique.

In this case, define ϕ = w + rx^{γ}. We refer to this case as the residue transcendental case. This case does not occur if \overline{κ} = ℝ.

2. Proof of Theorem 1.1.

We prove part (1) of Theorem 1.1. Every order embedding extends uniquely to the real closure so it suffices to prove that the extension of p to F′ given by y ↦ ϕ is an order embedding of F′ into \overline{κ′}((\overline{V′})). This is clear if y is algebraic. If y is transcendental, one uses the fact that orderings on \overline{F} (y) are completely determined by the associated cuts in \overline{F}; see [7]. The terminology we are using is the following:

2.1 Terminology. A lower cut in an ordered set (I, ≤) is a subset L of I such that i ≤ j and j ∈ L ⇒ i ∈ L. The associated upper cut in I is U = I\L. A cut in I is a pair (L, U) where L is a lower cut and U is the associated upper cut.

The lower cut in \overline{F} determined by the ordering on \overline{F} (y) is \{z ∈ \overline{F} | z < y\}. Consequently, to prove (1) we are reduced to showing that, for z ∈ \overline{F}, z < y iff p(z) < ϕ and z > y iff p(z) > ϕ. First suppose there exists z′ ∈ \overline{F} such that v(y − z) < v(y − z′). Then z′ − z = (y − z) − (y − z′) has the same sign as y − z. Of course, p(z′) − p(z) = p(z′ − z) has the same sign as z′ − z. Also, v(w − p(z′)) ≥ v(y − z′) and v(ϕ − w) ≥ v(y − z′) so v(ϕ − p(z′)) ≥ v(y − z′) > v(y −z) = v(z′ −z) = v(p(z′)−p(z)) so p(z′)−p(z) = (ϕ−p(z))−(ϕ−p(z′)) has the same sign as ϕ − p(z). Combining these things we see that ϕ − p(z) has the same sign as y − z as required. The other case is where S has a largest element γ and v(y − z) = γ. If γ ∉ \overline{V}, then p(z) = w+ terms of value > γ so, for all such z, either y > z and ϕ = w + x^{γ} > p(z) or y < z and ϕ = w − x^{γ} < p(z). Finally, if γ ∈ \overline{V} then ϕ = w + rx^{γ} and the result is clear from the definition of r.

We also need to check that the extended p is proper. This is a consequence of the following case-by-case analysis:

2.2 Note. (1) In the algebraic case, \overline{V′} = \overline{V}, \overline{κ′} = \overline{κ} and everything is clear. (2) In the immediate transcendental case, ϕ = w ∈ \overline{κ}((\overline{V})) so p(\overline{F′}) ⊆ \overline{κ}((\overline{V})). The induced embedding \overline{V′} ↪ \overline{V} is the identity on \overline{V}. Thus \overline{V′} = \overline{V}, the extended p is proper and \overline{κ′} = \overline{κ}. (3) In the residue transcendental case, ϕ = w + rx^{γ} so p(\overline{F′}) ⊆ \overline{κ(r)}((\overline{V})) and r ∈ \overline{κ′}. Thus \overline{V′} = \overline{V}, the extended p is proper and \overline{κ′} = \overline{κ(r)}. (4) In the value transcendental case, ϕ = w ± x^{γ} so p(\overline{F′}) ⊆ \overline{κ}((\overline{V} ⊕ ℚγ)). The induced embedding \overline{V′} ↪ \overline{V} ⊕ ℚγ is the identity on \overline{V} and sends γ to γ (since if z ∈ \overline{F} is such that v(y − z) = γ, then p(y − z) = ϕ − p(z) = ±x^{γ}+ terms of greater value, so the image of γ is v(p(y − z)) = γ). Thus \overline{V′} = \overline{V} ⊕ ℚγ, the extended p is proper in this case too and \overline{κ′} = \overline{κ}.

Thus the extension of p to \overline{F′} is order preserving and proper in all cases so the proof of part (1) of Theorem 1.1 is complete. □

In proving parts (2) and (3) of Theorem 1.1, there is no harm in identifying \overline{F} with p(\overline{F}) and y with ϕ. It is also convenient, for later work, to have the following intrinsic description of ϕ.

2.3 Definition. Suppose the embedding \overline{F} ⊆ \overline{κ}((\overline{V})) is proper. We say an element ϕ ∈ ℝ((H)), H some ordered group extension of \overline{V}, is a distinguished power series for \overline{F} if one of the following holds:

(1) ϕ ∈ \overline{F}.

(2) ϕ ∈ \overline{κ}((\overline{V})), ϕ ∉ \overline{F} and every proper truncation (ϕ)_{<γ} of ϕ is of the form (ϕ)_{<γ} = (c)_{<γ} for some c ∈ \overline{F}.

(3) ϕ = w ± x^{γ}, γ ∉ \overline{V} and w = (c)_{<γ} for some c ∈ \overline{F}.

(4) ϕ = w + rx^{γ} where r ∈ ℝ\\overline{κ}, γ ∈ \overline{V} and w = (c)_{<γ} for some c ∈ \overline{F}.

2.4 Theorem. Suppose \overline{F} ⊆ \overline{κ}((\overline{V})) is proper. If F′ = F (y) is an ordered extension of F then the canonically defined power series ϕ ∈ \overline{κ′}((\overline{V′})) corresponding to y is distinguished for \overline{F}. Conversely, if ψ ∈ ℝ((H)) is distinguished for \overline{F}, H some ordered group extension of \overline{V}, then there exists an ordered extension F′ = F (y) of F such that ψ is the canonically defined power series corresponding to y.

Proof. The first assertion is clear from Definition 1.3. For the second, take y = ψ and order F′ = F (y) via the embedding F′ ⊆ ℝ((H)). We complete the proof by checking that the canonically defined power series ϕ corresponding to y is equal to y. Let S = \{v(y − z) | z ∈ \overline{F} \}.

(1) Suppose y ∈ \overline{F}, i.e., ∞ ∈ S. Then clearly ϕ = y.

(2) Suppose y ∈ \overline{κ}((\overline{V})), y ∉ \overline{F}, but every proper truncation (y)_{<γ} of y is of the form (y)_{<γ} = (z)_{<γ} for some z ∈ \overline{F}. Then S has no largest element. (If v(y − z) = γ, z ∈ \overline{F}, then picking t ∈ \overline{F}, v(t) = γ, t = (y_{γ} − z_{γ})x^{γ}+ terms of higher value, we see that z′ = z + t ∈ \overline{F} satisfies v(y − z′) > γ.) If γ < v(y − z) for some z ∈ \overline{F} then, if γ ∉ \overline{V}, then y_{γ} = z_{γ} = 0 = w_{γ}, and, if γ ∈ \overline{V}, then y_{γ} = z_{γ} = w_{γ}. If γ > S and y_{γ} ≠ 0, then (y)_{<γ} is a proper truncation of y, so (y)_{<γ} = (z)_{<γ} for some z ∈ \overline{F} so v(y − z) ≥ γ, contradicting the definition of S. Thus y_{γ} = w_{γ} = 0 for γ > S. Thus ϕ = w = y.

(3) Suppose that y = w + ϵx^{γ}, ϵ = ±1, γ ∉ \overline{V}, and w = (z)_{<γ} for some z ∈ \overline{F}. Then v(y − z) ≥ γ. Also, for any z ∈ \overline{F}, z_{γ} = 0 ≠ y_{γ} = ±1, so v(y − z) ≤ γ.

Thus γ is the largest element of S. Also, for any z ∈ \overline{F} with v(y − z) = γ, z < y iff ϵ = 1 and z > y iff ϵ = −1. Thus ϕ = w + ϵx^{γ} = y.

(4) Suppose that y = w + rx^{γ}, r ∈ ℝ\κ, γ ∈ \overline{V}, and w = (z)_{<γ} for some z ∈ \overline{F}. Then again, γ is the largest element of S and, if z ∈ \overline{F}, v(y − z) = γ, then z < y iff z_{γ} < r and z > y iff z_{γ} > r. Thus ϕ = w + rx^{γ} = y. □

We return to the proof of Theorem 1.1. To simplify notation, identify \overline{F} with p(\overline{F}) and y with ϕ. From the way ϕ is defined we see that every proper truncation of ϕ is the truncation of an element of \overline{F}. If \overline{F} is truncation closed then every proper truncation of ϕ belongs to \overline{F} so, by [16, Lemma 3.4], K = \overline{F} (ϕ) is truncation closed. In particular, κ′ ⊆ K so the field L obtained by adjoining to K all elements of \overline{κ′} is algebraic over K. Also, any proper truncation of an element of \overline{κ′} is 0 ∈ K so, by repeated application of [16, Lemma 3.4], L is truncation closed. Thus x^{γ} ∈ L for all γ ∈ V′ so the field M obtained by adjoining to L all elements x^{γ/n}, γ ∈ V′, n ≥ 1, is algebraic over L and, as before, is truncation closed. M has value group \overline{V′} and residue field \overline{κ′} so, by [16, Lemma 3.5], \overline{M} is truncation closed. Since M is algebraic over K, \overline{M} = \overline{F′}. This completes the proof of part (2) of Theorem 1.1.

Suppose now that F ⊆ κ̃((V)). We want to show that ϕ ∈ κ̃′((V′)) or, equivalently, that F (ϕ) ⊆ κ̃′((V′)). Let F^{h} denote the henselization of F and let F (ϕ)^{h} denote the henselization of F (ϕ) [21]. We view F^{h} and F (ϕ)^{h} as valued subfields of the big algebraically closed valued field ℂ((\overline{V′})). Since F ⊆ κ̃((V)) and κ̃((V)) is henselian, F^{h} ⊆ κ̃((V)). Similarly, since F (ϕ) ⊆ \overline{κ′}((\overline{V′})) and \overline{κ′}((\overline{V′})) is henselian, F (ϕ)^{h} ⊆ \overline{κ′}((\overline{V′})). Also, F ⊆ F (ϕ) so F^{h} ⊆ F (ϕ)^{h}.

Claim 1. If c ∈ F (ϕ)^{h} is algebraic over F then c ∈ κ̃′((V′)). c is algebraic over F^{h} and F^{h} is henselian so, by [21, Th 2, p 236], the degree of F^{h}(c) over F^{h} is equal to the product of the ramification index and the residue degree. Choose d ∈ F^{h}(c) such that F^{h}(d) is unramified over F^{h} and F^{h}(c) is totally ramified over F^{h}(d). We can assume the minimal polynomial f of d over F^{h} is monic with coefficients in the valuation ring of F^{h}. Since F (ϕ)^{h} is an immediate extension of F (ϕ), the image of d in the residue field, call this \overline{d}, lies in κ′. Since F^{h} ⊆ κ̃((V)), the coefficients of f are also in κ̃((V)). By the existence assertion of Hensel’s Lemma there exists a root e of f in κ̃′((V)) such that \overline{e} = \overline{d}. By the uniqueness assertion of Hensel’s Lemma, e = d. Thus d ∈ κ̃′((V)) so F^{h}(d) ⊆ κ̃′((V)). Suppose γ is in the value group of F^{h}(c) and y ∈ F^{h}(c) is chosen so that v(y) = γ. Replacing y by −y, we can assume y > 0. There exists an integer n ≥ 1 and z ∈ F^{h}(d) such that y^{n} − z has value greater than nγ. Thus z = y^{n}(1 + t) where t ∈ F^{h}(c) has value > 0. F^{h}(c) is also henselian so (1 + t)^{1/n} ∈ F^{h}(c). Thus z^{1/n} = y(1 + t)^{1/n} ∈ F^{h}(c), and v(z^{1/n}) = γ. On the other hand, z ∈ F^{h}(d) ⊆ κ̃′((V)) so z = ax^{nγ}(1 + p) for some p ∈ κ̃′((V)) of positive value and a ∈ κ̃′, a > 0. Thus z^{1/n} = a^{1/n}x^{γ}(1 + p)^{1/n} ∈ κ̃′((V′)). Applying this to a set of generators γ_{1}, ..., γ_{k} of the value group of F^{h}(c) modulo V, we get elements z_{i} ∈ F^{h}(d) with z_{i}^{1/n_{i}} ∈ κ̃′((V′)), i = 1, ..., k, and F^{h}(c) = F^{h}(d, z_{1}^{1/n_{1}}, ..., z_{k}^{1/n_{k}}). This implies F^{h}(c) ⊆ κ̃′((V′)).

This completes the algebraic case. If ϕ ∈ \overline{F}, then ϕ is algebraic over F so by Claim 1, ϕ ∈ κ̃′((V′)). Suppose now that ϕ ∉ \overline{F}. To show ϕ ∈ κ̃′((V′)) in this case, it suffices to show the following: Claim 2. For each γ ∈ S, there exists c ∈ F (ϕ)^{h} algebraic over F with v(ϕ−c) ≥ γ.

For, by Claim 2 and the definition of w, each term w_{δ}x^{δ} appearing in w is a term of some c ∈ F (ϕ)^{h} which is algebraic over F so, by Claim 1, δ ∈ V′ and w_{δ} ∈ κ̃′. Thus w ∈ κ̃′((V′)). If S has no largest element then ϕ = w ∈ κ̃′((V′)). If S has a largest element γ then, by Claim 2, there exists c ∈ F (ϕ)^{h} algebraic over F such that v(ϕ − c) ≥ γ. Since c ∈ \overline{F}, the definition of S implies v(ϕ − c) = γ. (Recall: y and ϕ are identified now.) Since ϕ − c ∈ F (ϕ)^{h}, an immediate extension of F (ϕ), this implies γ ∈ V′. Thus, if γ ∉ \overline{V} then ϕ = w ± x^{γ} ∈ κ̃′((V′)). Suppose now that γ ∈ \overline{V} so ϕ = w + rx^{γ} for some r ∈ ℝ\\overline{κ}. Pick an integer n ≥ 1 such that nγ ∈ V and b ∈ F with v(b) = nγ. Then v(\frac{(ϕ−c)^{n}}{b}) = 0 and \frac{(ϕ−c)^{n}}{b} ∈ F (ϕ)^{h}, an immediate extension of F (ϕ), so the image of \frac{(ϕ−c)^{n}}{b} in the residue field belongs to κ′ and has the form u = \frac{(r−s)^{n}}{t} where s is the coefficient of x^{γ} in c and t is the coefficient of x^{nγ} in b. Thus t ∈ κ̃ and s ∈ κ̃′ (by Claim 1) so r ∈ κ̃′. Thus ϕ = w + rx^{γ} ∈ κ̃′((V′)) is this case too.

To prove Claim 2, fix γ ∈ S and d ∈ \overline{F} such that v(ϕ − d) = γ. Let H = F (ϕ)^{h} ∩ \overline{F}. H is an algebraic extension of F^{h} so is henselian. Let e and f denote the ramification index and residue degree of H(d) over H. Pick 1 = x_{1}, ..., x_{e} ∈ H(d) so that v(x_{1}), ..., v(x_{e}) generate the distinct cosets of the value group extension and 1 = y_{1}, ..., y_{f} ∈ H(d) of value zero such that the residues \overline{y_{1}}, ..., \overline{y_{f}} are a basis for the residue field extension. Thus ef is the degree of H(d) over H and the x_{i}y_{j}, i = 1, ..., e, j = 1, ..., f form a basis of H(d) over H. Also, by our construction, v(∑_{i,j} c_{ij}x_{i}y_{j}) = min_{i,j}{v(c_{ij}x_{i}y_{j})} for any c_{ij} ∈ H. In particular, if d = ∑_{i,j} c_{ij}x_{i}y_{j} then c = c_{11} is a ‘best approximation’ of d in H in the sense that v(d − c) ≥ v(d − c′) for any c′ ∈ H. It remains to prove that v(ϕ − c) ≥ γ. Suppose this is not the case. Say v(ϕ − c) = δ < γ. Then ϕ − c = qx^{δ} + ..., δ ∈ \overline{V}, q ≠ 0, q ∈ \overline{κ}. Then nδ ∈ V for some integer n ≥ 1 so we have some element a = tx^{nδ} + ... in F, t ∈ κ̃, t ≠ 0. Thus ϕ − c and a belong to F (ϕ)^{h} so (ϕ − c)^{n}/a ∈ F (ϕ)^{h}, v((ϕ − c)^{n}/a) = 0. The image of (ϕ − c)^{n}/a in the residue field is q^{n}/t. q and t belong to \overline{κ} so q^{n}/t is in \overline{κ} so is algebraic over the residue field κ of F. Thus, by Hensel’s Lemma applied to F (ϕ)^{h}, we have an element b in the valuation ring of F (ϕ)^{h}, algebraic over F and such that the image of b in the residue field is q^{n}/t. Thus (ϕ − c)^{n} − ab has value > nδ so ab = (ϕ−c)^{n}(1+p) where p ∈ F (ϕ)^{h} has value > 0. Then (1+p)^{1/n} ∈ F (ϕ)^{h} and ((ϕ−c)(1+p)^{1/n})^{n} = ab is algebraic over F, so (ϕ−c)(1+p)^{1/n} is algebraic over F. Take c′ = c+(ϕ−c)(1+p)^{1/n}. Thus v(ϕ−c′) > v(ϕ−c) and v(ϕ−d) > v(ϕ−c) so v(d − c′) > v(ϕ − c) = v(d − c). Since c′ ∈ H, this contradicts the definition of c. □

3. Orderings on the polynomial ring \overline{κ}[Y_{1}, ..., Y_{ℓ}].

Let A be a commutative ring with 1. By an ordering on A we mean a subset P of A such that P + P ⊆ P, P P ⊆ P, P ∪ −P = A, and 𝔭 := P ∩ −P (called the support of P) is a prime ideal of A. Equivalently, an ordering on A is a pair (𝔭, ≤) where 𝔭 is a prime ideal of A and ≤ is an ordering on qf(A/𝔭), the quotient field of the domain A/𝔭 [4] [5] [10] [15].

Consider the set-up of Theorem 1.1. In the algebraic, immediate transcendental and residue transcendental cases, ϕ is an element of ℝ((\overline{V})). In the value transcendental case, ϕ has the form ϕ = w +ϵx^{γ} where ϵ = ±1, γ ∉ \overline{V} belongs to some ordered group extension of \overline{V} and w ∈ \overline{κ}((\overline{V})) is obtained from an element of \overline{F} by truncating the terms with value ≥ γ. Unfortunately, the value group \overline{V′} = \overline{V} ⊕ ℚγ is only determined up to isomorphism over \overline{V}. If γ_{1} ∉ \overline{V} is an element in some ordered group extension of \overline{V} defining the same cut in \overline{V} as the element γ, then γ ↦ γ_{1} defines an order isomorphism \overline{V} ⊕ ℚγ ≅ \overline{V} ⊕ ℚγ_{1} over \overline{V} [23]. Thus, if we use this other group as the value group, then the power series we obtain changes from ϕ = w + ϵx^{γ} to ϕ_{1} = w + ϵx^{γ_{1}}. This is not very satisfactory. To get a canonical power series ϕ associated to each value transcendental ordering, we need a canonical way of choosing the elements γ we use to fill the various cuts in \overline{V}.

For (I, ≤) a totally ordered set, let H = H(I) denote the Hahn group of rank I. By definition, H is the ordered Abelian group consisting of all functions α : I → ℝ having well-ordered support with pointwise addition, ordered lexicographically. Denote by v_{s} : H → I ∪ {∞} the natural set valuation, i.e., v_{s}(0) = ∞ and, for α ∈ H, α ≠ 0, v_{s}(α) is the least i such that α(i) ≠ 0 [6].

Let \overline{I} denote the set of all cuts of I. The set I′ = I · ∪ \overline{I} has a natural ordering so we can form the Hahn group H′ = H(I′). The natural embedding I ↪ I′ is order preserving and induces an ordered group embedding H ↪ H′ preserving the set valuation v_{s}. Here we are mainly interested in the case where I is finite. Note: if |I| = n then |\overline{I}| = n + 1 so |I′| = 2n + 1. Thus, if we iterate the process, starting with I = ∅, we obtain a sequence of Hahn groups

ℍ_{0} ↪ ℍ_{1} ↪ ...

defined recursively by ℍ_{0} = H(∅) = {0}, ℍ_{i+1} = (ℍ_{i})′. We can identify ℍ_{i} with the product of 2^{i} − 1 copies of (ℝ, +, ≤), ordered lexicographically. If we do this, the embedding ℍ_{i} ↪ ℍ_{i+1} is given by

(r_{1}, ..., r_{2^{i}−1}) ↦ (0, r_{1}, 0, ..., 0, r_{2^{i}−1}, 0).

3.1 Lemma. Let \overline{V} be a (divisible) subgroup of ℍ_{i}. Then each cut (S_{1}, S_{2}) in \overline{V} is filled by a canonically defined element γ in ℍ_{i+1}.

Proof. The definition of γ is analogous to the definition of ϕ given in Section 1. Let

T = {v_{s}(α_{2} − α_{1}) | α_{1} ∈ S_{1}, α_{2} ∈ S_{2}}.

There is a unique element δ in ℍ_{i} such that δ(j) = 0 if j > v_{s}(α_{2} − α_{1}) for all α_{1} ∈ S_{1} and all α_{2} ∈ S_{2}, δ(j) = α_{1}(j) if j < v_{s}(α_{2} − α_{1}) for some α_{1} ∈ S_{1} and some α_{2} ∈ S_{2} and, if j is the largest element of T, then α_{1}(j) ≤ δ(j) ≤ α_{2}(j) for all α_{1} ∈ S_{1} and all α_{2} ∈ S_{2} satisfying v_{s}(α_{1} − α_{2}) = j. Note: if T = ∅, i.e., if one of S_{1}, S_{2} is empty, then δ = 0. If there does not exist an element α ∈ \overline{V} such that

δ(i) = \{ \begin{matrix}α(i) & if i ∈ T \\ 0 & if i ∉ T \end{matrix} ,

we take γ = δ. If such an element α ∈ \overline{V} exists, then either all such α belong to S_{1} or all such α belong to S_{2}. In this case, we take γ = δ ± χ_{{n}}, with the ‘+’ sign if all these α belong to S_{1}, and the ‘−’ sign if they all belong to S_{2}, where χ_{{n}} denotes the characteristic function of {n} and n ∈ v_{s}(ℍ_{i+1}) is chosen to fill the gap between the last element of T and the first element of v_{s}(\overline{V}) greater than the last element of T. (Note: This makes sense even if ∞ is the only element of v_{s}(\overline{V}) greater than the last element of T. It also makes sense if T = ∅.) There may be several such n. To get a canonical γ, look at the smallest k, 1 ≤ k ≤ i+1 such that there is such an element n in v_{s}(ℍ_{k}). Then, in v_{s}(ℍ_{k}) there is exactly one such element n, and this is the one we pick. □

Fix an archimedean ordered field κ. Using Lemma 3.1, we will see now that orderings on the polynomial ring \overline{κ}[Y_{1}, ..., Y_{ℓ}] are in natural one-to-one correspondence with certain distinguished ℓ-tuples of the form (ϕ_{1}, ..., ϕ_{ℓ}) where ϕ_{i} ∈ ℝ((ℍ_{i})), i = 1, ..., ℓ.

These are defined recursively. Suppose that (ϕ_{1}, ..., ϕ_{j}) has been defined, 0 ≤ j < ℓ, F_{j} ⊆ κ̃_{j}((V_{j})) and \overline{F_{j}} is truncation closed in \overline{κ_{j}}((\overline{V_{j}})), where F_{j} = κ(ϕ_{1}, ..., ϕ_{j}), V_{j} ⊆ ℍ_{j} denotes the value group of F_{j} and κ_{j} denotes the residue field of F_{j}. Then ϕ_{j+1} is any element of the set ∪_{i=1}^{4}S_{i} ⊆ ℝ((ℍ_{j+1})) where:

• S_{1} = \overline{F_{j}}.

• S_{2} consists of all ϕ ∈ \overline{κ_{j}}((\overline{V_{j}})) such that ϕ is not element of \overline{F_{j}} but every proper truncation of ϕ is an element of \overline{F_{j}}.

• S_{3} consists of all elements ϕ = w ± x^{γ}, γ the distinguished element of ℍ_{j+1} corresponding to some cut in \overline{V_{j}} (see Lemma 3.1), w an element of \overline{F_{j}} having no terms of value ≥ γ.

• S_{4} consists of all elements ϕ = w + rx^{γ}, γ ∈ \overline{V_{j}}, r ∈ ℝ\\overline{κ_{j}}, w an element of \overline{F_{j}} having no terms of value ≥ γ.

In this way, (ϕ_{1}, ..., ϕ_{j+1}) is defined. Note: By Theorem 2.4 and parts (2) and (3) of Theorem 1.1, ϕ_{j+1} ∈ { \atop κ_{j+1}}((V_{j+1})) and \overline{F_{j+1}} is truncation closed in \overline{κ_{j+1}}((\overline{V_{j+1}})).

3.2 Remark. The sets S_{1}, S_{2}, S_{3}, S_{4} correspond to the algebraic, immediate transcendental, value transcendental and residue transcendental cases respectively. If \overline{κ} = ℝ then κ_{j} = κ̃_{j} = \overline{κ_{j}} = ℝ and the set S_{4} corresponding to the residue transcendental case is the empty set.

The ordering on \overline{κ}[Y_{1}, ..., Y_{ℓ}] corresponding to the ℓ-tuple (ϕ_{1}, ..., ϕ_{ℓ}) is defined to be the one induced by the \overline{κ}-algebra homomorphism \overline{κ}[Y_{1}, ..., Y_{ℓ}] → ℝ((ℍ_{ℓ})), Y_{i} ↦ ϕ_{i}, i = 1, ..., ℓ.

3.3 Theorem. The correspondence defined above, between distinguished ℓtuples and orderings on \overline{κ}[Y_{1}, ..., Y_{ℓ}], is one-to-one and onto.

Proof. Let P_{j} = P ∩\overline{κ}[Y_{1}, ..., Y_{j}], 0 ≤ j ≤ ℓ where P is an ordering of \overline{κ}[Y_{1}, ..., Y_{ℓ}]. Suppose, by induction, that there is a unique distinguished j-tuple (ϕ_{1}, ..., ϕ_{j}) defining P_{j}, 0 ≤ j < ℓ. Let F_{j} = κ(ϕ_{1}, ..., ϕ_{j}) ⊆ κ̃_{j}((V_{j})). Extensions of P_{j} to \overline{κ}[Y_{1}, ..., Y_{j+1}] are in one-to-one correspondence with orderings on \overline{F_{j}}[Y_{j+1}]. Orderings of \overline{F_{j}}[Y_{j+1}] having support ≠ \{0\} correspond to elements of S_{1}. By Theorem 1.1 (1), Theorem 2.4 and Lemma 3.1, support \{0\} orderings of \overline{F_{j}}[Y_{j+1}] correspond to elements of S_{2} ∪ S_{3} ∪ S_{4}. Here, S_{1}, S_{2}, S_{3}, S_{4} are defined as above. Thus we have a unique distinguished j + 1-tuple (ϕ_{1}, ..., ϕ_{j+1}) defining P_{j+1}. □

3.4 Notes. (1) The support of the ordering is the kernel of the corresponding mapping from \overline{κ}[Y_{1}, ..., Y_{ℓ}] to ℝ((ℍ_{ℓ})), call this 𝔭_{ℓ}. trdeg(F_{ℓ} : κ) is the depth of 𝔭_{ℓ} and ℓ− trdeg(F_{ℓ} : κ) is the height of 𝔭_{ℓ}. ℓ− trdeg(F_{ℓ} : κ) also counts the number of algebraic ϕ_{i} occuring in (ϕ_{1}, ..., ϕ_{ℓ}). The following well-known inequality holds:

trdeg(F_{ℓ} : κ) ≥ dim_{ℚ}(\overline{V_{ℓ}}) + trdeg(κ_{ℓ} : κ).

dim_{ℚ}(\overline{V_{ℓ}}) counts the number of value transcendental ϕ_{i}. trdeg(κ_{ℓ} : κ) counts the number of residue transcendental ϕ_{i}. The difference

trdeg(F_{ℓ} : κ) − (dim_{ℚ}(\overline{V_{ℓ}}) + trdeg(κ_{i} : κ))

counts the number of immediate transcendental ϕ_{i}. These assertions are clear from Note 2.2.

(2) Orderings on \overline{κ}(Y_{1}, ..., Y_{ℓ}), i.e., support zero orderings on \overline{κ}[Y_{1}, ..., Y_{ℓ}], correspond to distinguished ℓ-tuples (ϕ_{1}, ..., ϕ_{ℓ}) where all of the ϕ_{i} are transcendental.

(3) The value group V_{ℓ} is generated by the union of the supports of the ϕ_{i}, i = 1, ..., ℓ. κ̃_{ℓ} is the smallest field containing κ and the coefficients of the ϕ_{i} and closed under taking n-th roots of positive elements, n ≥ 1.

(4) Every ℓ-tuple (ψ_{1}, ..., ψ_{ℓ}), ψ_{i} ∈ ℝ((ℍ_{ℓ})), i = 1, ..., ℓ, gives rise to a \overline{κ}- homomorphism \overline{κ}[Y_{1}, ..., Y_{ℓ}] → ℝ((ℍ_{ℓ})) via Y_{i} ↦ ψ_{i}, i = 1, ..., ℓ, and hence to an ordering on \overline{κ}[Y_{1}, ..., Y_{ℓ}]. Let us view two such ℓ-tuples as being equivalent if they define the same ordering on \overline{κ}[Y_{1}, ..., Y_{ℓ}]. By Theorem 3.3, for any such ℓ-tuple (ψ_{1}, ..., ψ_{ℓ}), there is a unique distinguished ℓ-tuple in the same class. Unfortunately, it seems very difficult to describe this distinguished ℓ-tuple in any simple way in terms of ψ_{1}, ..., ψ_{ℓ}. In this sense, the problem poised by MacLane and Schilling in [11, Final Remark] remains unsolved.

(5) For example, the specializations of the ordering [4, p. 35] [5, p. 116] corresponding to a given distinguished ℓ-tuple (ϕ_{1}, ..., ϕ_{ℓ}) are naturally represented by (not necessarily distinguished) ℓ-tuples (ϕ′_{1}, ..., ϕ′_{ℓ}) where each ϕ′_{i} is a suitable truncation of ϕ_{i}, i = 1, ..., ℓ, but it is difficult to describe explicitly the distinguished ℓ-tuples corresponding to these specializations.

3.5 Example. Orderings on \overline{κ}[Y]. Here, ℍ_{0} = \{0\}, ℍ_{1} = ℝ. In the algebraic case, ϕ = a, a ∈ \overline{κ}. The immediate transcendental case does not occur. In the value transcendental case, either ϕ = ±x^{−1} (w = 0, γ = −1) or ϕ = a ± x, a ∈ \overline{κ} (w = a, γ = 1). In the residue transcendental case, ϕ = a, a ∈ ℝ\\overline{κ} (w = 0, γ = 0, r = a).

4. Real places on \overline{κ}[Y_{1}, ..., Y_{ℓ}].

Define a 2-character of V_{ℓ} to be a group homomorphism σ : V_{ℓ} → {−1, 1}. Every 2-character σ of V_{ℓ} induces an ℝ-automorphism of the field ℝ((V_{ℓ})) given by f = ∑ a_{γ}x^{γ} ↦ f_{σ} = ∑ a_{γ}σ(γ)x^{γ}.

4.1 Remark. Every 2-character on V_{ℓ} factors through the group V_{ℓ}/2V_{ℓ}. The group V_{ℓ}/2V_{ℓ} is finite. If α_{1} + 2V_{ℓ}, ..., α_{n} + 2V_{ℓ} are ℤ/(2)-independent, then α_{1}, ..., α_{n} are ℚ-independent. Thus |V_{ℓ}/2V_{ℓ}| ≤ 2^{dim_{ℚ}(\overline{V_{ℓ}})}. Consequently, the number of 2-characters of V_{ℓ} is equal to the order of the group V_{ℓ}/2V_{ℓ}.

Let A be a commutative ring with 1. A real place on A is defined to be a pair (𝔭, λ) where 𝔭 is a prime ideal of A and λ : qf(A/𝔭) → ℝ ∪ {∞} is a place. The real place associated to an ordering (𝔭, ≤) of A is (𝔭, λ) where λ is the real place on qf(A/𝔭) associated to the ordering ≤; see [10] [15] [19].

4.2 Corollary. If (ϕ_{1}, ..., ϕ_{ℓ}) is a distinguished ℓ-tuple then, for each 2-character σ of V_{ℓ}, (ϕ_{1σ}, ..., ϕ_{ℓσ}) is a distinguished ℓ-tuple. The map σ ↦ (ϕ_{1σ}, ..., ϕ_{ℓσ}) is injective. Orderings corresponding to the distinguished ℓ-tuples of the form (ϕ_{1σ}, ..., ϕ_{ℓσ}), σ a 2-character of V_{ℓ}, all belong to the same real place of \overline{κ}[Y_{1}, ..., Y_{ℓ}] and are the full equivalence class of orderings belonging to this real place.

Proof. We prove, by induction on j, that (ϕ_{1σ}, ..., ϕ_{jσ}) is a distinguished j-tuple, j = 0, ..., ℓ. Suppose 0 ≤ j < ℓ, F_{j} = \overline{κ}(ϕ_{1}, ..., ϕ_{j}), F_{jσ} = \overline{κ}(ϕ_{1σ}, ..., ϕ_{jσ}). Let V_{j} (resp., κ_{j}) denote the value group (resp., residue field) of F_{j}. The automorphism f ↦ f_{σ} is valuation preserving and maps F_{j} onto F_{jσ} so V_{j} (resp., κ_{j}) is the value group (resp., residue field) of F_{jσ}. There are four cases. (1) ϕ_{j+1} is algebraic over F_{j}. Then ϕ_{(j+1)σ} is algebraic over F_{jσ}. (2) ϕ_{j+1} ∈ \overline{κ_{j}}((\overline{V_{j}})) is not algebraic over F_{j} but any proper truncation of ϕ_{j+1} is algebraic over F_{j}. Then ϕ_{(j+1)σ} ∈ \overline{κ_{j}}((\overline{V_{j}})) is not algebraic over F_{jσ}, but any proper truncation of ϕ_{(j+1)σ} is algebraic over F_{jσ}. (3) ϕ_{j+1} = w ± x^{γ}, w algebraic over F_{j} with no terms of value ≥ γ, and γ is the distinguished element in ℍ_{j+1} corresponding to some cut in \overline{V_{j}}. Then ϕ_{(j+1)σ} = w_{σ} ± σ(γ)x^{γ}, w_{σ} is algebraic over F_{jσ} with no terms of value ≥ γ, and γ is the distinguished element in ℍ_{j+1} corresponding to some cut in \overline{V_{j}}. (4) ϕ_{j+1} = w + rx^{γ}, w algebraic over F_{j} having no terms of value ≥ γ, γ ∈ \overline{V_{j}}, r ∈ ℝ\\overline{κ_{j}}. Then ϕ_{(j+1)σ} = w_{σ} + rσ(γ)x^{γ}, w_{σ} is algebraic over F_{jσ} having no terms of value ≥ γ, γ ∈ \overline{V_{j}}, and rσ(γ) = ±r ∈ ℝ\\overline{κ_{j}}. Thus, in any case, (ϕ_{1σ}, ..., ϕ_{(j+1)σ}) is a distinguished j + 1-tuple. The result follows by induction on j.

The real place associated to (ϕ_{1}, ..., ϕ_{ℓ}) is clearly the pair (𝔭_{ℓ}, λ), where 𝔭_{ℓ} is the kernel of the \overline{κ}-homomorphism from \overline{κ}[Y_{1}, ..., Y_{ℓ}] to F_{ℓ}, Y_{i} ↦ ϕ_{i}, i = 1, ..., ℓ, and λ is the composite place

qf(\frac{\overline{κ}[Y_{1}, ..., Y_{ℓ}]}{𝔭_{ℓ}}) ≅ F_{ℓ} ⊆ ℝ((V_{ℓ})) ⟶^{μ} ℝ ∪ \{∞\}

where the isomorphism is the \overline{κ}-isomorphism induced by Y_{i} ↦ ϕ_{i}, i = 1, ..., ℓ and μ is the canonical place on ℝ((V_{ℓ})). Since μ(f_{σ}) = μ(f) for any f ∈ ℝ((V_{ℓ})), this is the same as the place associated to (ϕ_{1σ}, ..., ϕ_{ℓσ}).

V_{ℓ} is generated by the elements γ such that x^{γ} appears in some ϕ_{i} so, if σ, τ are distinct 2-characters of V_{ℓ} then ϕ_{jσ} ≠ ϕ_{jτ} for some j so (ϕ_{1σ}, ..., ϕ_{ℓσ}) ≠ (ϕ_{1τ} , ..., ϕ_{ℓτ}). The last assertion follows by counting, using the Baer-Krull Theorem, see [15, Theorem 1.3.2]. □

4.3 Examples. (1) Real places on \overline{κ}(Y). In the value transcendental case, V_{1} = ℤ so there are exactly two 2-characters of V_{1}. (Either σ(1) = 1 or σ(1) = −1.) ϕ = a + x and ϕ = a − x have the same associated real place. Also ϕ = x^{−1} and ϕ = −x^{−1} have the same associated real place. In the residue transcendental case, V_{1} = {0}, a divisible group, so the only 2-character of V_{1} is the trivial one.

(2) For the ordering on \overline{κ}(Y_{1}, Y_{2}) corresponding to

ϕ_{1} = x, ϕ_{2} = ∑_{i=1}^{∞} x^{1−1/i},

V_{1} = ℤ, V_{2} = ℚ. Since V_{2} is divisible, the only 2-character of V_{2} is the trivial one.

4.4 Remark. The results in Sections 3 and 4 carry over, in a suitably generalized form, to orderings on the polynomial ring \overline{F} [Y_{1}, ..., Y_{ℓ}], F an arbitrary ordered field; see [26].

5. Building extensions with prescribed value group.

We assume F′ = F (y), an ordered extension of F. We choose an embedding p : \overline{F} → \overline{κ}((\overline{V})) as in Corollary 1.2, and identify F with its image in κ̃((V)). Our next result seems to be well-known. In any case, parts (1), (2) and (3) are well-known.

5.1 Theorem. V′ and κ′ are restricted as follows:

(1) Algebraic case: V′/V is finite and κ′ is a finite extension of κ.

(2) Value transcendental case: V′ = ℤδ ⊕ W where ℤδ is infinite cyclic, W ⊇ V, W/V is finite, κ′ is a finite extension of κ.

(3) Residue transcendental case: V′/V is finite, κ′ is a rational function field in one variable over a finite extension of κ.

(4) Immediate transcendental case: V′/V is countable torsion, \overline{κ}((V)) ⊈ \overline{F} if V′/V is finite, κ′ is an algebraic extension of κ.

Proof. (1) is standard ([24], Corollary 2, p. 26, and Corollary, p. 52).

(2) Pick b ∈ F′ so that v(b) ∉ \overline{V}. Then F′ is a finite extension of F (b), the value group of F (b) is V ⊕ ℤγ, γ = v(b), and the residue field of F (b) is κ. (2) is now clear, using (1).

(3) Pick b ∈ F′, v(b) = 0, so that the image of b in the residue field is transcendental over κ. Then F′ is a finite extension of F (b) and the value group of F (b) is V; hence as in (1), V′/V is finite. By the Ruled Residue Theorem [17], κ′ is a rational function field in one variable over a finite extension of κ.

(4) κ′ ⊆ \overline{κ} so κ′ is algebraic over κ. For each γ ∈ \overline{V} such that x^{γ} appears in ϕ, there exists z ∈ \overline{F} with v(ϕ − z) > γ. Thus if W_{γ} = the group generated over V by the set of δ ∈ \overline{V} such that δ ≤ γ and x^{δ} appears in ϕ, then W_{γ}/V is finite. This implies that V′/V is the union of an increasing chain of finite groups, so is countable. If V′/V is finite and \overline{κ}((V)) ⊆ \overline{F} then \overline{κ}((V′)) ⊆ \overline{F}, contradicting ϕ ∉ \overline{F}. □

5.2 Theorem. Modulo the restrictions imposed by Theorem 5.1, the value group V′ can be prescribed arbitrarily.

Proof. In cases (1) (2) (3) below, W denotes an ordered group extension of V with W/V finite (so W ⊆ \overline{V}).

(1) Algebraic case. Suppose W/V ≅ ⊕_{i=1}^{s}ℤ/(e_{i}). Let β_{i} = e_{i}α_{i}, i = 1, ..., s where α_{1}, ..., α_{s} are elements of W corresponding to the natural generators of ⊕_{i=1}^{s}ℤ/(e_{i}), and let F′ = F ( \sqrt[e_{1}]{p_{1}}, ..., \sqrt[es]{p_{s}}) where p_{i} ∈ F is such that v(p_{i}) = β_{i}, p_{i} > 0. Use the primitive element theorem to pick q ∈ \overline{F} so that F′ = F (q). By our construction, V′ = W.

(2) Value transcendental case. Suppose U = ℤδ ⊕ W, an ordered Abelian extension of W with ℤδ infinite cyclic. Pick q ∈ \overline{F} so that the value group of F (q) is W (as in Case 1). By Theorem 1.1, the exponents γ such that x^{γ} appears in q generate W over V. Since W/V is finite there is a element μ ∈ W such that the exponents γ such that x^{γ} appears in q and γ ≤ μ generate W over V. Replacing δ by −δ if necessary, we can assume δ > 0. Replacing δ by δ + μ if necessary, we can assume δ > μ. Let ϕ = w + x^{δ} where w is obtained from q by truncating the terms in x^{γ} with γ > δ. Then F′ = F (ϕ) is value transcendental over F and, by Theorem 1.1 and Theorem 2.4, the value group of F′ is V′ = ℤδ ⊕ W = U.

(3) Residue transcendental case. Pick q ∈ \overline{F} so that the value group of F (q) is W (as in Case 1). Choose μ as in Case 2 and let ϕ = w+rx^{μ} where w is obtained from q by truncating the terms in x^{γ} with γ ≥ μ and r ∈ ℝ\\overline{κ} (if such r exists). Then F′ = F (ϕ) is residue transcendental over F and V′ = W by Theorem 1.1 and Theorem 2.4.

(4) Immediate transcendental case. Suppose first that V ⊆ W ⊆ \overline{V} and W/V is finite. Pick q ∈ \overline{F} so the value group of F (q) is W. Pick q′ ∈ \overline{κ}((V))\\overline{F} (if such an element exists). Multiplying q′ by an element of F of sufficiently large value, we can assume the exponents γ such that x^{γ} appears in q and γ < δ generate W over V, where δ = v(q′). Take S = the set of elements γ ∈ \overline{V} such that there exists a ∈ \overline{F} such that v(q +q′ −a) = γ and let ϕ be obtained from q +q′ by truncating the terms with value not in S. Then F′ = F (ϕ) is immediate transcendental, and clearly V′ = W by Theorem 1.1 and Theorem 2.4. Next suppose V ⊆ W ⊆ \overline{V} and that W/V is countably infinite. Pick generators α_{1}, α_{2}, ... for W over V. We can assume each α_{i} is positive. Let β_{1} = α_{1}, β_{2} = α_{1} + α_{2}, ... Then β_{1} < β_{2} < ... and these elements generate W over V. Pick q_{i} ∈ \overline{F} so that the value group of F (q_{i}) is contained in W and v(q_{i}) = β_{i} (using Case 1). Let p_{1} = q_{1}, i_{1} = 1. Not all of the β_{i} are in the value group of F (p_{1}). Let i_{2} be the first i such that β_{i} is not in the value group of F (p_{1}), and let p_{2} = p_{1} + q_{i_{2}}. Similarly, let i_{3} be the first i such that β_{i} is not in the value group of F (p_{2}) and let p_{3} = p_{2} + q_{i_{3}}, etc.. In this way we get a sequence of elements p_{1}, p_{2}, ... in \overline{F}. Let S be the set of elements of \overline{V} which are < β_{i_{k}} for some k (so the sequence β_{i_{1}} < β_{i_{2}} < ... is cofinal in S and S has no largest element). Also take ϕ to be the power series in \overline{κ}((\overline{V})) such that ϕ has no term in x^{γ} if γ ∉ S, and such that v(ϕ − p_{j}) ≥ β_{i_{j}}. Then F′ = F (ϕ) is immediate transcendental over F and V′ = W. □

5.3 Remark. It can be shown that also the residue field extension can be prescribed arbitrarily, subject to the restriction imposed by Theorem 5.1; cf. [12].

6. Characterization of ℵ_{0}–saturated real closed fields.

Let an ordinal α be given. Recall that a totally ordered set Δ is an η_{α}– set, if given Δ_{1}, Δ_{2} subsets of Δ of cardinality < ℵ_{α} and such that Δ_{1} < Δ_{2} then there exists δ ∈ Δ such that Δ_{1} < δ < Δ_{2}. For example Δ is a dense linear orderings without endpoints if and only if Δ is an η_{0}–set. Also, since the theory of dense linear orderings without endpoints admits elimination of quantifiers [22], it follows that a dense linear ordering without endpoints is ℵ_{α}– saturated (as ordered set) if and only if it is an η_{α}–set. In [1], a beautiful valuation theoretic characterization of ordered Abelian groups and ordered fields which are η_{α}–sets for a nonzero α is given. Divisible ordered Abelian groups (respectively real closed fields) are o–minimal [18]. Thus for ℵ_{α} > ℵ_{0}, they are ℵ_{α}–saturated as ordered groups (respectively ordered fields) if and only if they are ℵ_{α}–saturated as ordered sets, that is, if and only if they are η_{α}–sets [13]. Thus one obtains easily from Alling’s characterization a characterization of the ℵ_{α}–saturated divisible ordered Abelian groups and real closed fields, for uncountable ℵ_{α}. This characterization however does not cover the case of the ℵ_{0}–saturated models. In [13] this case was studied for divisible ordered Abelian groups:

6.1 Theorem. Let G ≠ 0 be a divisible ordered Abelian group. Then G is ℵ_{0}–saturated (in the language of ordered groups) if and only if its value set is a dense linear ordering without endpoints and all its archimedean components are isomorphic to ℝ.

We consider here the case of real closed fields. Below, let R denote a real closed field, v its natural valuation, G its value group and ρ its residue field. We prove the following

6.2 Theorem. Let ℵ_{α} ≥ ℵ_{0}, then R is ℵ_{α}–saturated (in the language of ordered fields) if and only if

1) G is ℵ_{α}–saturated

2) ρ ≃ ℝ

3) every pseudo Cauchy sequence in a subfield of absolute transcendence degree less than ℵ_{α} has a pseudo limit in R.

Before proving the theorem let us note the following. Since R is real closed, G is divisible, and thus, by elimination of quantifiers [22], is o-minimal as noted above. More precisely, a formula ϕ(x, g_{1}, ..., g_{n}) in one free variable x and parameters g_{1}, ..., g_{n} ∈ G is equivalent to a finite union of intervals with endpoints in ⟨g_{1}, …, g_{n}⟩∪{−∞, +∞} (for A ⊂ G, ⟨A⟩ denotes the smallest divisible subgroup of G containing A). Thus if q is a complete 1-type over G with parameters in A ⊂ G, to realize q it is necessary and sufficient to realize the following set of formulas (determined by q):

{g ≤ x; g ∈ ⟨A⟩ , q ⊢ g ≤ x} ∪ {x ≤ h ; h ∈ ⟨A⟩ , q ⊢ x ≤ h}.

Similarly, by elimination of quantifiers [22], R is o-minimal. That is, a formula ϕ(x, a_{1}, …, a_{n}) with parameters in A = {a_{1}, …, a_{n}} ⊂ R, is equivalent to a finite union of intervals with endpoints in \overline{ℚ(A)}∪\{−∞, +∞\} (here \overline{ℚ(A)} denotes the relative algebraic closure of ℚ(A) in R.) Thus if q is a complete 1-type over R with parameters in A ⊂ R, to realize q it is necessary and sufficient to realize the set:

\{b ≤ x; b ∈ \overline{ℚ(A)} , q ⊢ b ≤ x\} ∪ \{x ≤ c; c ∈ \overline{ℚ(A)} , q ⊢ x ≤ c\} .

Now we prove the theorem. Suppose that 1), 2), 3) are satisfied, we show that R is ℵ_{α}–saturated. By 2), we identify ρ with ℝ.

Let A ⊂ R with card A < ℵ_{α} and q a 1-type over R with parameters in A (w.l.o.g., assume that q is a complete type). We want to realize q in R. By the remark preceeding the proof, it is enough to realize in R the set

q′(x) = \{b ≤ x; b ∈ \overline{ℚ(A)} , q ⊢ b ≤ x\} ∪ \{x ≤ c; c ∈ \overline{ℚ(A)} , q ⊢ x ≤ c\} .

If q′(x) contains an equality, the result is obvious. So suppose that we only have strict inequalities in q′(x).

Set R′ = \overline{ℚ(A)}, B = \{b ∈ R′; q ⊢ b < x\}, C = \{c ∈ R′; q ⊢ x < c\}. Let R′′ be an elementary extension of R in which q is realized; and x_{0} ∈ R′′ s.t. R′′ ⊧ q(x_{0}). Consider the following subset of v(R′′):

Δ = {v(d − x_{0}) ; d ∈ R′} .

There are three cases to consider.

Immediate transcendental case: Δ has no largest element. Thus,

∀d ∈ R′ ∃d′ ∈ R′ : v(d′ − x_{0}) > v(d − x_{0}).

Let {v(d_{λ} − x_{0})}_{λ<μ} be cofinal in Δ, then {d_{λ}}_{λ<μ} is a pseudo Cauchy sequence in R′ and trdeg (R′|ℚ) < ℵ_{α}. Let a ∈ R be a pseudo limit. We claim that a realizes q′(x); indeed we have:

v(a − x_{0}) = v(a − d_{λ} + d_{λ} − x_{0}) ≥ min{v(a − d_{λ}), v(d_{λ} − x_{0})}

and

v(a − d_{λ}) = v(d_{λ+1} − d_{λ}) = v(x_{0} − d_{λ}) ,

thus for all λ we have v(a − x_{0}) ≥ v(d_{λ} − x_{0}); thus for all d ∈ R′ we have v(a − x_{0}) > v(d − x_{0}).

Let b ∈ B; we show that b < a. If not, b ≥ a, so a ≤ b < x_{0}, which implies that v(b − x_{0}) ≥ v(a − x_{0}), contradiction. Similarly one shows that if c ∈ C, then a < c.

Value transcendental case: Assume now that Δ has a largest element γ ∉ v(R′), and fix d_{0} ∈ R′ s.t. v(d_{0} − x_{0}) = γ is the maximum of Δ. Note that if d_{0} ∈ B then for all d ∈ R′ with v(d − x_{0}) = γ we must have that d ∈ B, and if d_{0} ∈ C then for all d ∈ R′ with v(d − x_{0}) = γ we must have that d ∈ C. Assume first that d_{0} ∈ B (the case d_{0} ∈ C is treated similarly). Consider Δ_{1} = {v(c − d_{0}) ; c ∈ C} and Δ_{2} = {v(b − d_{0}) ; b ∈ B, b > d_{0}}, we claim that Δ_{1} < γ < Δ_{2}. Indeed since d_{0} ∈ B, we must have v(c − x_{0}) < γ for all c ∈ C, thus

v(c − d_{0}) = v(c − x_{0} + x_{0} − d_{0}) = min{v(c − x_{0}), v(x_{0} − d_{0})} = v(c − x_{0}) < γ.

If b ∈ B , b ≥ d_{0} then v(x_{0} − b) ≥ v(x_{0} − d_{0}) = γ and equality must hold since γ is the largest element. Thus,

v(b − d_{0}) = v(b − x_{0} + x_{0} − d_{0}) ≥ min{v(b − x_{0}), v(x_{0} − d_{0})} = γ.

Since γ ∉ v(R′) we must have v(b − d_{0}) > γ. This establishes our claim. Hence,

t(y) = {v(c − d_{0}) < y; c ∈ C} · ∪ {y < v(b − d_{0}) ; b ∈ B , b > d_{0}}

is a type in G with parameters in G′ = v(R′). If ℵ_{α} > ℵ_{0} then card G′ < ℵ_{α} and we can realize t(y) in G by condition 1) of the theorem. If ℵ_{α} = ℵ_{0}, then R′ is a real closed field of finite absolute transcendence degree, so G′ has finite rational rank [24; § 10]. Thus if {g_{1}, …, g_{n}} is a ℚ–basis of G′, then t(y) is in fact a type in the parameters {g_{1}, …, g_{n}}, hence can be realized by some g ∈ G (since G is ℵ_{0}–saturated). Let a ∈ R, a > 0 s.t. v(a) = g, then (by definition of t(y))

∀c ∈ C ∀b ∈ B : b > d_{0} ⟹ v(c − d_{0}) < v(a) < v(b − d_{0}) ,

hence (by compatibility of v with the order)

∀c ∈ C ∀b ∈ B : b > d_{0} ⟹ b − d_{0} < a < c − d_{0} ,

i.e.,

∀c ∈ C ∀b ∈ B : b > d_{0} ⟹ b < a + d_{0} < c ,

which implies

∀c ∈ C ∀b ∈ B : b < a + d_{0} < c ,

so a + d_{0} ∈ R realizes q′(x) as required.

Residue transcendental case: Assume now that the largest element γ of Δ is in v(R′), and fix a > 0, a ∈ R′ and d_{0} ∈ R′ s.t. v(d_{0} − x_{0}) = γ = v(a). Claim: there exist b_{0} ∈ B and c_{0} ∈ C s.t. for all b ∈ B with b ≥ b_{0} and for all c ∈ C with c ≤ c_{0} we have

v(b − d_{0}) = γ = v(a) = v(c − d_{0})

(note that then for such b’s and c’s we will have also that v(b−x_{0}) = v(x_{0}−d_{0}) = γ = v(c − x_{0})). Indeed, since v(d_{0} − x_{0}) = v(a) there exists n ∈ ℕ s.t. na > |x_{0} − d_{0}| > a/n. If d_{0} ∈ B (i.e. d_{0} < x_{0}) we set c_{0} = d_{0} + na and b_{0} = d_{0} + a/n. Clearly c_{0} > x_{0} so c_{0} ∈ C and b_{0} < x_{0} so b_{0} ∈ B. Moreover v(b_{0} − d_{0}) = v(a/n) = v(a) = v(na) = v(c_{0} − d_{0}). For b ∈ B, b > b_{0} and c ∈ C, c < c_{0} we have d_{0} < b_{0} < b < c < c_{0}. Thus v(b − d_{0}) ≤ v(b_{0} − d_{0}) = γ and γ = v(c_{0} − d_{0}) ≤ v(b − d_{0}) hence equality must hold. Similarly one shows that γ = v(c − d_{0}). If d_{0} ∈ C (i.e. d_{0} > x_{0}) we set b_{0} = d_{0} − na and c_{0} = d_{0} − a/n. Clearly c_{0} > x_{0} so c_{0} ∈ C and b_{0} < x_{0} so b_{0} ∈ B. Moreover v(b_{0} − d_{0}) = v(−na) = v(a) = v(−a/n) = v(c_{0} − d_{0}). For b ∈ B, b > b_{0} and c ∈ C, c < c_{0} we have b_{0} < b < c < c_{0} < d_{0}. Thus γ = v(b_{0} − d_{0}) ≤ v(b − d_{0}) and γ = v(c_{0} − d_{0}) ≥ v(b − d_{0}), thus equality must hold. Similarly one shows that γ = v(c − d_{0}). This establishes the claim.

Consider

\{\frac{b − d_{0}}{a} < x′′; b ∈ B , b ≥ b_{0}\} ∪ \{x′′ < \frac{c − d_{0}}{a} ; c ∈ C , c ≤ c_{0}\} .

If x′′ ∈ R realizes this, then x′ = x′′ · a ∈ R realizes

{b − d_{0} < x′; b ∈ B , b ≥ b_{0}} ∪ {x′ < c − d_{0} ; c ∈ C , c ≤ c_{0}}

thus x′ + d_{0} ∈ R realizes q′(x). Thus we just need to find x′′. Assume that d_{0} ∈ B (the case d_{0} ∈ C is treated similarly). Note that by the claim for all b ∈ B with b ≥ b_{0} and for all c ∈ C with c ≤ c_{0} we have

v (\frac{b − d_{0}}{a} ) = v (\frac{x_{0} − d_{0}}{a} ) = v (\frac{c − d_{0}}{a} ) = 0 ,

and taking residues,

\overline{\frac{b − d_{0}}{a}} < \overline{\frac{x_{0} − d_{0}}{a}} < \overline{\frac{c − d_{0}}{a}}

(note that the inequalities are strict, since otherwise we get a contradiction to the maximality of v(a) in Δ). Consider the cut in the reals

\{\overline{\frac{b − d_{0}}{a}} ; b ∈ B , b ≥ b_{0}\} · ∪ \{\overline{\frac{c − d_{0}}{a}} ; c ∈ C , c ≤ c_{0}\} .

By the above strict inequalities, r′′ := \overline{\frac{x_{0}−d_{0}}{a}} =∈ ℝ satisfies that

\{\overline{\frac{b − d_{0}}{a}} ; b ∈ B , b ≥ b_{0}\} < r′′ < \{\overline{\frac{c − d_{0}}{a}} ; c ∈ C , c ≤ c_{0}\} .

Let x′′ ∈ R s.t. \overline{x′′} = r′′ (x′′ exists by condition 2) of the theorem). This completes the residue transcendental case, and the only if direction of the proof.

Now suppose that R is ℵ_{α}–saturated. We show that 1), 2), 3) must hold.

1): Let q(x) be a type in the parameters {g_{α}; α ∈ λ} ⊂ G with λ < ℵ_{α} (w.l.o.g. assume q is a complete type). Set H = ⟨{g_{α}; α ∈ λ}⟩. It suffices to realize in G the set

{g ≤ x; g ∈ H , q(x) ⊢ g ≤ x} ∪ {x ≤ g; g ∈ H , q(x) ⊢ x ≤ g} .

If the set contains an equality, the result is clear. So suppose that we only have strict inequalities.

For every α ∈ λ, fix a_{α} ∈ R, a_{α} > 0 s.t. v(a_{α}) = g_{α}. If g ∈ H and g = q_{1}g_{α_{1}} + … + q_{n}g_{α_{n}} with q_{1}, …, q_{n} ∈ ℚ, then g = v(a_{α_{1}}^{q_{1}} · . . . · a_{α_{n}}^{q_{n}}) where for simplicity we agree to choose a_{α_{i}}^{q_{i}} > 0 for all i ∈ {1, …, n}. Let

H_{1} = {g ∈ H; q(x) ⊢ g < x} and H_{2} = {g ∈ H; q(x) ⊢ x < g}

and consider

q′(x) = {na_{α_{1}}^{q_{1}} · . . . · a_{α_{n}}^{q_{n}} < x; n ∈ ℕ , v(a_{α_{1}}^{q_{1}} · . . . · a_{α_{n}}^{q_{n}}) ∈ H_{2}}
· ∪ {mx < a_{α_{1}}^{q_{1}} · . . . · a_{α_{n}}^{q_{n}} ; m ∈ ℕ , v(a_{α_{1}}^{q_{1}} · . . . · a_{α_{n}}^{q_{n}}) ∈ H_{1}} .

Since R is a dense linear ordering without endpoints, q′(x) is finitely realizable in R. Thus q′(x) is a type in the parameters {a_{α}; α ∈ λ} with λ < ℵ_{α}. Hence q′(x) is realized in R, say by a. It is clear that v(a) realizes q(x).

2): By hypothesis, (R, +, 0, <) is ℵ_{0}–saturated, so by Theorem 3.1, all archimedean components are isomorphic to ℝ. In particular, ρ ≃ ℝ.

3): Let (a_{ν})_{ν<μ} be a pseudo Cauchy sequence in a subfield R_{0} of R s.t. trdeg (R_{0}|ℚ) = λ < ℵ_{α}. Let {b_{α}; α ∈ λ} be a transcendence basis of R_{0} over ℚ. Then every element of R_{0} is definable by a formula with parameters in {b_{α}; α ∈ λ}. Consider

q_{1}(x) = {n|x − a_{ν+1}| < |a_{ν} − a_{ν+1}|; ν < μ , n ∈ ℕ} ;

then q_{1}(x) is a set of formulas in the parameters {b_{α}; α ∈ λ}. Moreover, q_{1}(x) is finitely satisfied in R since (a_{ν})_{ν<μ} is pseudo Cauchy. Hence q_{1}(x) is a type, and it is clear that a realization of q_{1}(x) in R is a pseudo limit of the sequence. This completes the proof of the theorem.

6.3 Corollary. If R is a maximally valued real closed field, then R is not saturated.

Proof. Let card R = ℵ_{α}, R saturated and maximally valued. Let G = v(R), then R ≃ ℝ((G)). By the theorem, G is also ℵ_{α}–saturated and thus contains a well ordered subset of cardinality ℵ_{α}. It follows that card ℝ((G)) ≥ 2^{ℵ_{α}} > ℵ_{α}, contradiction. □

References

1. N.L. Alling, On the existence of real closed fields that are η_{α}–sets of power ℵ_{α}, Trans. Amer. Math. Soc. 103 (1962), 341–352.

2. M.E. Alonso, Ordenes y clausura semientera, PhD. Thesis, Univ. Complutense de Madrid, 1984.

3. M.E. Alonso, J.M. Gamboa, J.M. Ruiz, On orderings in real surfaces, J. Pure Appl. Algebra 36 (1985), 1–14.

4. C. Andradas, L. Bröcker, J.M. Ruiz, Constructible sets in real geometry, Ergeb. Math., vol. 33, Springer-Verlag, 1996.

5. J. Bochnak, M. Coste, M-F. Roy, Géométrie Algébrique Réelle, Ergeb. Math., Springer-Verlag, 1987.

6. L. Fuchs, Partially ordered algebraic systems, Pergamon Press, New York, 1963.

7. R. Gilmer, Extension of an order to a simple transcendental extension, in Ordered fields and real algebraic geometry (San Francisco, 1981), eds. D. Dubois and T. Récio, Cont. Math., vol. 8, Amer. Math. Soc., 1982, pp. 113–118.

8. H. Hahn, Über die nichtarchimedischen Grössensysteme, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Vienna, Section IIa, 116 (1907), 601–653.

9. I. Kaplansky, Maximal fields with valuations, Duke Math. J. 9 (1942), 303–321.

10. M. Knebusch, C. Scheiderer, Einführung in reelle Algebra, Aufbaukurs Math., Vieweg, Braunschweig Wiesbaden, 1989.

11. W. Krull, Allgemeine Bewertungstheorie, J. für Math. 167 (1932), 160–196.

12. F.-V. Kuhlmann, Valuations on Rational Function Fields, I: Implicit Constant Fields, in preparation.

13. S. Kuhlmann, Groupes abéliens divisibles ordonnés, Publications mathématiques de l’Université Paris VII, Séminaire sur les Structures Algébriques Ordonnées, Sélection d’exposés 1984–1987, Vol. 1 (1990), 3–14.

14. S. MacLane, O.F.G. Schilling, Zero dimensional branches of rank one on algebraic varieties, Annals of Math. 40 (1939), 507–520.

15. M. Marshall, Spaces of orderings and abstract real spectra, Lecture Notes in Math., vol. 1636, Springer, 1996.

16. M.H. Mourgues, J.P. Ressayre, Every real closed field has an integer part, J. Symb. Logic 58 (1993), 641–647.

17. J. Ohm, The Ruled Residue Theorem for simple transcendental extensions of valued fields, Proc. Amer. Math. Soc. 89 (1983), 16–18.

18. A. Pillay, C. Steinhorn, Definable sets in ordered structures I, Trans. Amer. Math. Soc. 295 (1986), 565–592.

19. A. Prestel, Lectures on formally real fields, IMPA, Rio de Janeiro, 1974; Lecture Notes in Math., vol. 1093, Springer, 1984.

20. S. Prieß-Crampe, Angeordnete Strukturen. Gruppen, Körper, projektive Ebenen, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 98, Springer, 1983.

21. P. Ribenboim, Théorie des valuations, Les Presses de L’Université de Montreal, 1968.

22. A. Robinson, Complete Theories, Amsterdam, 1956.

23. R. Rolland, Étude des coupures dans les groupes et corps ordonnés, in Géométrie algébrique réelle et formes quadratiques (Rennes, 1981), eds. J-L. Colliot-Thélène, M. Coste, L. Mahé, M-F. Roy, Lecture Notes in Mathematics, vol. 959, Springer, 1982, pp. 386–405.

24. P. Samuel, O. Zariski, Commutative Algebra, Vol. II, New York–Heidelberg–Berlin, 1960.

25. O.F.G. Schilling, The theory of valuations, Math. Surveys, vol. 4, Amer. Math. Soc., 1950.

26. M. Zekavat, Orderings, cuts and formal power series, PhD. Thesis, Univ. of Saskatchewan, in preparation.

Department of Mathematics & Statistics, University of Saskatchewan, Saskatoon, SK Canada, S7N 5E6

E-mail address: fvk, skuhlman, marshall, zekavat@math.usask.ca