Images of additive polynomials in 𝔽_{q}((t)) have the optimal approximation property^{∗}

Lou van den Dries and Franz-Viktor Kuhlmann

16. 1. 2001

Abstract We show that the set of values of an additive polynomial in several variables with arguments in a formal Laurent series field over a finite field has the optimal approximation property: every element in the field has a (not necessarily unique) closest approximation in this set of values. The approximation is with respect to the canonical valuation on the field. This property is elementary in the language of valued rings.

1 Introduction

Let 𝔽_{q} denote the field with q elements, where q is a power of a prime p. The power series field 𝔽_{q}((t)), also called “field of formal Laurent series over 𝔽_{q}”, carries a canonical valuation v_{t}, the t-adic valuation, with value group ℤ and v_{t}(t) = 1. In studying elementary properties of this valued field the following notion turns up, see [K].

Let (K, v) be a valued field, and S a nonempty subset of K. We say that S has the optimal approximation property (OA) in (K, v) if for every point in K there is a (not necessarily unique) closest point in S, that is, for every x ∈ K there exists y ∈ S such that

v(x − y) = max{v(x − z) | z ∈ S} .

(We write valuations in the additive Krull style, that is, the ultrametric triangle law reads as v(a + b) ≥ min{va, vb}. Thus elements a, b are close if v(a − b) is large. We denote the value group of (K, v) by vK.) The following implications hold (see Section 2):

S compact ⇒ S has OA ⇒ S is closed.

∗While working on this paper, the first author was partially supported by NSF grant DMS 98-02745.

The second author was partially supported by a Canadian NSERC grant. Both authors would like to thank the MSRI for support and hospitality.

AMS Subject Classification: 12J10, 12L12, 03C60

This approximation property relates to the model theory of valued fields since it is elementary for (elementarily) definable S. The image

S := {f(a_{1}, …, a_{n}) | a_{1}, …, a_{n} ∈ K}  (1)

of K under a polynomial f ∈ K[X_{1}, …, X_{n}] is definable, so the question arises when this image has OA. In [K] the second author shows that the image of an algebraically maximal field under a polynomial in one variable has OA. (A valued field is said to be algebraically maximal if it has no proper algebraic valued field extension preserving both value group and residue field.)

We consider here the case of additive polynomials in several variables. Let K be a field of characteristic p > 0. A polynomial f(X_{1}, …, X_{n}) ∈ K[X_{1}, …, X_{n}] is called additive if

f(a_{1} + b_{1}, …, a_{n} + b_{n}) = f(a_{1}, …, a_{n}) + f(b_{1}, …, b_{n})

for all elements a_{1}, …, a_{n}, b_{1}, …, b_{n} in any extension field of K. If f is additive, then

f(X_{1}, …, X_{n}) = f_{1}(X_{1}) + … + f_{n}(X_{n})  (2)

where each f_{i}(X_{i}) := f(0, …, 0, X_{i}, 0 …, 0) is an additive polynomial in one variable. We refer to [L, VIII, §11] for the fact that the additive polynomials in one variable X over K are precisely the polynomials of the form

∑_{i=0}^{m} c_{i}X^{p^{i}} with c_{i} ∈ K , m ∈ ℕ.

A valued field is called maximal if it has no proper valued field extension preserving both value group and residue field. In [K] it is shown that if (K, v) is maximal, then under a certain additional (elementary) condition on the additive polynomial f in several variables, the image S has OA. It would be desirable to remove this additional condition. We do this here for (𝔽_{q}((t)), v_{t}), using its local compactness:

Theorem 1 If f is an additive polynomial in several variables with coefficients in 𝔽_{q}((t)), then the image of 𝔽_{q}((t)) under f has the optimal approximation property in (𝔽_{q}((t)), v_{t}).

It would be nice to generalize this to all maximal valued fields, in other words, to replace the use of local compactness by some other argument.

The axiom scheme which by Theorem 1 holds in (𝔽_{q}((t)), v_{t}) consists of the sentences

∀(c_{i,j}) ∀x ∃y_{1} … ∃y_{n} ∀z_{1} … ∀z_{n} : v(x − ∑_{i=1}^{n} ∑_{j=0}^{n} c_{i,j}y_{i}^{p^{j}}) ≥ v(x − ∑_{i=1}^{n} ∑_{j=0}^{n} c_{i,j}z_{i}^{p^{j}})

in the language of valued rings. In [K] this scheme is shown to be independent of the following more familiar axioms satisfied by (𝔽_{q}((t)), v_{t}): “henselian defectless (= algebraically complete) valued field with value group a ℤ-group and residue field 𝔽_{q}”. We suspect that the elementary theory of the valued field (𝔽_{q}((t)), v_{t}) is completely axiomatizable by augmenting these familiar axioms with sentences that express just properties of additive polynomials (like those provided by Theorem 1).

To be more explicit about this suspicion, let us briefly review a (well-known) moduletheoretic interpretation of additive polynomials. Suppose the field K is infinite and of characteristic p > 0. Then the endomorphism ring of the additive group of K has subring K[φ], where λ ∈ K ⊆ K[φ] acts on K as multiplication by λ, and φ acts as the Frobenius map x ↦ x^{p}; so φλ = λ^{p}φ in K[φ]. This makes K a left module over K[φ], and the images of K under the additive polynomials in several variables over K are exactly the additive subgroups of the K[ϕ]-module K of the form f_{1}K + … + f_{n}K, where f_{1}, …, f_{n} ∈ K[ϕ]. (Since K[ϕ] is not commutative, these additive subgroups are not in general submodules of K.) For (K, v) = (𝔽_{q}((t)), v_{t}), its elementary theory as a valued module over K[φ] has yet to be determined in a satisfactory way; our theorem is exactly about this valued module. A complete description of the elementary theory of this valued module seems essential in reaching an understanding of the elementary theory of the valued field (K, v).

The second author would like to thank Hervé Perdry for discussions on Remark 1 at the end of this paper, and Trevor Green for proof-reading.

2 Compactness and OA

Let (K, v) be a non-trivially valued field, α ∈ vK and a ∈ K. The closed ball B_{α}(a) and the open ball B_{α}^{∘}(a) are defined as follows:

B_{α}(a) = {b ∈ K | v(a − b) ≥ α} and B_{α}^{∘}(a) = {b ∈ K | v(a − b) > α} .

Both kinds of balls are open and closed in the topology induced by the valuation, and are easily seen to have OA in (K, v). (But they are not compact if (K, v) is not locally compact.) Note that if vK is dense, then K \ B_{α}(0) is closed but does not have OA in (K, v) since it contains no closest point to 0.

Lemma 2 Suppose S is a nonempty compact subset of K. Then S has OA in (K, v).

Proof: Let x ∈ K \ S. If z ∈ S, then z ∉ B_{v(x−z)}^{∘}(x), so the collection

{K \ B_{v(x−z)}^{∘}(x) | z ∈ S}

is an open covering of S, hence contains a finite subcovering. The collection of balls B_{v(x−z)}^{∘}(x) is totally ordered by inclusion. It follows that the finite subcovering contains a largest set, say, K\B_{v(x−y)}^{∘}(x), which consequently contains S. That is, S∩B_{v(x−y)}^{∘}(x) = ∅. Since y ∈ S, this means that v(x − y) = max{v(x − z) | z ∈ S}. □

This proof actually provides a characterization of the optimal approximation property: A nonempty set S ⊆ K has OA in (K, v) if and only if every covering of S by the complements of a system of open balls with common center has a finite subcovering.

From the continuity of polynomial maps and Lemma 2 we conclude the following: If (K, v) is locally compact, then for every f ∈ K[X_{1}, …, X_{n}], α ∈ vK and a ∈ K, the image {f(a_{1}, …, a_{n}) | a_{1}, …, a_{n} ∈ B_{α}(a)} of B_{α}(a) has OA in (K, v).

We also have the following easy implication for nonempty S ⊆ K:

S has OA in (K, v) ⟹ S is closed.

As noted above, the converse fails if vK is dense. However, by a standard argument, this converse does hold if (K, v) is locally compact. (The validity of this converse for definable S amounts to a further set of elementary properties of the valued field (𝔽_{q}((t)), v_{t}).)

3 Valuation independence

Let (K, v) be a valued field, L a subfield of K, and (b_{i})_{i∈I} a system of non-zero elements in K, with I ≠ ∅. We call this system L-valuation independent if for every choice of elements a_{i} ∈ L such that a_{i} ≠ 0 for only finitely many i ∈ I, we have

v (∑_{i∈I} a_{i}b_{i}) = min_{i∈I} v(a_{i}b_{i}).

If V is an L-subvector space of K, then this system is called a valuation basis of V if it is a basis of V and L-valuation independent.

Let d = p^{ν}, ν ∈ ℕ, (K, v) = (𝔽_{q}((t)), v_{t}) and L = K^{d} := {a^{d} | a ∈ K} = 𝔽_{q}((t^{d})). Note that 1, t, t^{2}, …, t^{d−1} is a valuation basis of K as vector space over L.

Let V be an L-subvector space of K with basis b_{1}, …, b_{m}. We now indicate how to modify this basis to a valuation basis of V. Write b_{i} = ∑_{j=0}^{d−1} c_{ij}t^{j} with c_{ij} ∈ L. Take j_{1} to be the unique index such that vb_{1} = vc_{1j_{1}}t^{j_{1}}. Replacing b_{1} by b_{1}/c_{1j_{1}} we may assume c_{1j_{1}} = 1. Next, for every i ≥ 2, replace b_{i} by b_{i}−c_{ij_{1}}b_{1}, so we reduce to the case that c_{ij_{1}} = 0 for i ≥ 2. Repeat this procedure with the new elements b_{2}, …, b_{m}. By construction, the coefficients of t^{j_{1}} in the representations of these elements are zero. Thus, if vb_{2} = vc_{2j_{2}}t^{j_{2}} (where c_{2j_{2}} denotes the new coefficient), then j_{2} ≠ j_{1} . Hence, applying the procedure a total of m times we obtain a new basis of V which by an abuse of language we also call b_{1}, …, b_{m}, such that vb_{1}, …, vb_{m} are distinct elements of {0, 1, …, d − 1}. In particular, this new basis is a valuation basis of V.

4 Proof of Theorem 1

Throughout this section (K, v) = (𝔽_{q}((t)), v_{t}). Let

S := {f(a_{1}, …, a_{n}) | a_{1}, …, a_{n} ∈ K},

be the image of K under some additive polynomial f ∈ K[X_{1}, …, X_{n}].

We choose a system of non-zero additive polynomials h_{1}, …, h_{k} ∈ K[X] in one variable such that

h_{1}(K) + … + h_{k}(K) = S  (3)

for which ∑_{i=1}^{k} deg h_{i} is minimal. This is possible by (2). The idea is to modify this system to one that makes it so to say “visible” that S has OA in (K, v).

Denote by c_{i} the leading coefficient and by d_{i} the degree of h_{i}, for 1 ≤ i ≤ k. Then:

Lemma 3 For every choice of a_{1}, …, a_{k} ∈ K, not all zero, we have

∑_{i=1}^{k} c_{i}a_{i}^{d_{i}} ≠ 0 .

Proof: Suppose there are a_{1}, …, a_{k} ∈ K, not all zero, such that ∑_{i=1}^{k} c_{i}a_{i}^{d_{i}} = 0. After renumbering we may assume a_{1} ≠ 0 and d_{1} = max{d_{i} | 1 ≤ i ≤ k and a_{i} ≠ 0}. Replacing every a_{i} by a_{i}a_{1}^{−d_{1}/d_{i}}, we may even assume that a_{1} = 1. Now we set

h̃_{1}(X) := ∑_{i=1}^{k} h_{i}(a_{i}X^{d_{1}/d_{i}}) .

Since each polynomial h_{i}(a_{i}X^{d_{1}/d_{i}}) has degree d_{1} and leading coefficient c_{i}a_{i}^{d_{i}}, and since ∑_{i=1}^{k} c_{i}a_{i}^{d_{i}} = 0, we obtain deg h̃_{1} < d_{1} = deg h_{1} . Therefore,

deg h̃_{1} + ∑_{i=2}^{k} deg h_{i} < ∑_{i=1}^{k} deg h_{i} .  (4)

On the other hand, for every choice of b_{i} ∈ K we have

h_{1}(b_{1}) + … + h_{k}(b_{k}) = ∑_{i=1}^{k} h_{i}(a_{i}b_{1}^{d_{1}/d_{i}}) + ∑_{i=2}^{k} (h_{i}(b_{i}) − h_{i}(a_{i}b_{1}^{d_{1}/d_{i}}))

= h̃_{1}(b_{1}) + ∑_{i=2}^{k} h_{i}(b_{i} − a_{i}b_{1}^{d_{1}/d_{i}}) .

Thus S ⊆ h̃_{1}(K)+h_{2}(K)+…+h_{k}(K). The converse inclusion follows from the definition of h̃_{1} and the fact that S is an additive subgroup of K which contains the images h_{i}(K) for all i. So

S = h̃_{1}(K) + h_{2}(K) + … + h_{k}(K) ,

which in view of (4) contradicts the minimality of the system h_{1}, …, h_{k}. □

Lemma 4 There are additive polynomials g_{1}, …, g_{m} ∈ K[X] in one variable such that a) S = g_{1}(K) + … + g_{m}(K),

b) all polynomials g_{i} have the same degree d = p^{ν}, for some non-negative integer ν,

c) the leading coefficients b_{1}, …, b_{m} of g_{1}, …, g_{m} are such that vb_{1}, …, vb_{m} are distinct elements of {0, 1, …, d − 1}.

Proof: We set

d := max_{i} d_{i} and δ_{i} := d/d_{i} .

Since the h_{i} are additive polynomials, these numbers are powers of p. Hence

K = K^{δ_{i}} + tK^{δ_{i}} + … + t^{δ_{i}−1}K^{δ_{i}} .

Therefore,

h_{i}(K) = h_{i}(K^{δ_{i}}) + h_{i}(tK^{δ_{i}}) + … + h_{i}(t^{δ_{i}−1}K^{δ_{i}}) = h_{i,0}(K) + … + h_{i,δ_{i}−1}(K)

where

h_{i,j}(X) := h_{i}(t^{j}X^{δ_{i}}) ∈ K[X] .

Consequently,

S = ∑_{i=1}^{k} ∑_{j=0}^{δ_{i}−1} h_{i,j}(K)

with all polynomials h_{i,j} having degree d.

We claim that the leading coefficients c_{ij} = c_{i}t^{jd_{i}} of the polynomials h_{i,j} are K^{d}-linearly independent. Suppose that for a_{ij} ∈ K,

0 = ∑_{i=1}^{k} ∑_{j=0}^{δ_{i}−1} c_{ij} a_{ij}^{d} = ∑_{i=1}^{k} c_{i} ∑_{j=0}^{δ_{i}−1} t^{jd_{i}}a_{ij}^{δ_{i}d_{i}} = ∑_{i=1}^{k} c_{i} ( ∑_{j=0}^{δ_{i}−1} t^{j}a_{ij}^{δ_{i}} )^{d_{i}} .

Lemma 3 then gives

∑_{j=0}^{δ_{i}−1} t^{j}a_{ij}^{δ_{i}} = 0 for 1 ≤ i ≤ k .

As 1, t, …, t^{δ_{i}−1} are K^{δ_{i}}-linearly independent, it follows that a_{ij} = 0 for all i and j. This proves our claim.

We have now found additive polynomials h̃_{1}, …, h̃_{m} in K[X] of degree d, with K^{d}- linearly independent leading coefficients c̃_{1}, …, c̃_{m} and such that S = h̃_{1}(K) + … + h̃_{m}(K). The previous section shows that the K^{d}-vector space generated by c̃_{1}, …, c̃_{m} admits a valuation basis b_{1}, …, b_{m}, say, for which vb_{1}, …, vb_{m} are distinct elements of {0, 1, …, d − 1}. Write b_{i} = ∑_{j=1}^{m} r_{ij}^{d}c̃_{j} with r_{ij} ∈ K. Now we set

g_{i}(X) := ∑_{j=1}^{m} h̃_{j}(r_{ij}X)

and observe that for each i the polynomial g_{i} is of degree d with leading coefficient b_{i}. It only remains to show that condition a) is satisfied. Since S is an additive subgroup of K and contains the images h̃_{j}(K) for all j it follows that g_{1}(K) + … + g_{m}(K) ⊆ h̃_{1}(K)+…+h̃_{m}(K) = S. On the other hand, both c̃_{1}, …, c̃_{m} and b_{1}, …, b_{m} are bases, so the matrix (r_{ij}^{d}) is invertible. Thus, also the matrix (r_{ij}) is invertible. Denote its inverse by (s_{ij}), with s_{ij} ∈ K. A simple computation then shows that h̃_{i} = ∑_{j=1}^{m} g_{j}(s_{ij}X). Hence S = h̃_{1}(K) + … + h̃_{m}(K) ⊆ g_{1}(K) + … + g_{m}(K), which concludes the proof. □

Lemma 5 Suppose the additive polynomials g_{1}, …, g_{m} ∈ K[X] satisfy conditions b) and c) of Lemma 4. Then there exists α ∈ vK = ℤ such that if B is the additive subgroup B_{α}(0) of K and C a group complement of B in K, then for all b ∈ g_{1}(B) + … + g_{m}(B) and all non-zero c ∈ g_{1}(C) + … + g_{m}(C),

vc < dα ≤ vb .

Proof: Let f(X) = c_{n}X^{n} + … + c_{1}X + c_{0} ∈ K[X] be any polynomial, c_{n} ≠ 0. Take α ∈ vK such that if a ∈ K and va ≤ α then

vc_{n}a^{n} = vc_{n} + nva < vc_{j} + jva = vc_{j}a^{j} for 0 ≤ j < n ,

which implies

vf(a) = min_{0≤j≤n} vc_{j}a^{j} = vc_{n}a^{n} = vc_{n} + nva .  (5)

This in turn implies that for a ∈ K with va ≥ α we have

vf(a) ≥ min_{0≤j≤n} vc_{j}a^{j} ≥ vc_{n} + nα .  (6)

Now choose α such that (5) holds simultaneously for all f = g_{i}, 1 ≤ i ≤ m, and all a ∈ K with va ≤ α. Hence (6) holds simultaneously for all f = g_{i} and all a ∈ K with va ≥ α. As before, denote by b_{i} the leading coefficient of g_{i}. Then every a ∈ K with va ≤ α satisfies vg_{i}(a) = vb_{i} + dva. Since vb_{1}, …, vb_{m} are distinct elements of {0, 1, …, d − 1}, we find that for all choices of a_{i} ∈ K with va_{i} < α for at least one i, we have vb_{i} + dva_{i} < dα for this i, and

v(g_{1}(a_{1}) + … + g_{m}(a_{m})) = min_{i} vg_{i}(a_{i}) = min_{i} vb_{i} + dva_{i} < dα

by (5). But if va_{i} ≥ α for all i, then by (6),

v(g_{1}(a_{1}) + … + g_{m}(a_{m})) ≥ min_{i} vg_{i}(a_{i}) ≥ min_{i} vb_{i} + dα ≥ dα .

Let B and C be as in the lemma. Then B ∩ C = {0}, so every non-zero c ∈ C satisfies vc < α. Now the lemma follows from the inequalities above. □

Lemma 6 Let (F, v) be a valued field with value group vF = ℤ. Suppose B and C are non-trivial additive subgroups of F such that B has OA in (F, v) and vc < vb for all b ∈ B and all non-zero c ∈ C. Then B + C has OA in (F, v).

Proof: Take any x ∈ F. Since B ≠ {0}, the set {vc | 0 ≠ c ∈ C} is bounded from above in vF = ℤ, so vC has a maximum γ.

Suppose first that v(x − z) ≤ γ for all z ∈ C. Then {v(x − z) | z ∈ C} has a maximum in ℤ. Also, if b ∈ B and c ∈ C, then v(x − c) ≤ γ < vb, and therefore,

v(x − (b + c)) = min{v(x − c), vb} = v(x − c) ≤ max{v(x − z) | z ∈ C} ,

showing that {v(x − z) | z ∈ B + C} has a maximum, namely max{v(x − z) | z ∈ C}.

Now assume that v(x − c_{0}) > γ, with c_{0} ∈ C. Our assumption on B implies that the set {v(x − c_{0} − z) | z ∈ B} has a maximum, say, v(x − c_{0} − b_{0}) with b_{0} ∈ B. Note that v(x − c_{0} − b_{0}) > γ. Take any b ∈ B and c ∈ C. Then

v(x − (b + c)) ≥ min{v(x − c_{0} − b_{0}), v(c_{0} − c), v(b_{0} − b)} .

If c_{0} ≠ c, then v(c_{0} − c) < v(b_{0} − b) and v(c_{0} − c) ≤ γ < v(x − c_{0} − b_{0}), showing that v(x−(b+c)) = v(c_{0}−c) < v(x−c_{0}−b_{0}). If c = c_{0} , then v(x−(b+c)) ≤ v(x−c_{0}−b_{0}) holds by our choice of b_{0} . Thus {v(x−z) | z ∈ B+C} has a maximum, namely v(x−c_{0}−b_{0}). □

Proof of Theorem 1: Let f be an additive polynomial in several variables with coefficients in K = 𝔽_{q}((t)). Write the image of K under f as S = g_{1}(K) + … + g_{m}(K) with additive polynomials g_{1}, …, g_{m} ∈ K[X] in one variable which satisfy conditions b) and c) of Lemma 4. We choose α, B and C as in Lemma 5. Since B and C are additive subgroups of K, the additivity of the g_{i} implies that B := g_{1}(B) + … + g_{m}(B) and C := g_{1}(C) + … + g_{m}(C) are again additive subgroups of K. Moreover, as B is compact, so is B, and thus B has OA. Lemma 5 implies that the hypothesis of Lemma 6 is satisfied, so B + C has OA. But by the additivity of the g_{i},

B + C = g_{1}(B + C) + … + g_{m}(B + C) = g_{1}(K) + … + g_{m}(K) = S .

This concludes our proof. □

Remark 1. Theorem 1 goes through when (𝔽_{q}((t)), v_{t}) is replaced by any henselian valued subfield (L, v_{t}|_{L}) of (𝔽_{q}((t)), v_{t}) such that 𝔽_{q}(t) ⊂ L and [L : L^{p}] = p.

To see this, note that by Greenberg’s approximation theorem for discrete henselian valuation rings ([G]), such a subfield (L, v_{t}|_{L}) is existentially closed in (𝔽_{q}((t)), v_{t}) (the conditions 𝔽_{q}(t) ⊂ L and [L : L^{p}] = p imply that 𝔽_{q}((t))|L is separable). Let f be an additive polynomial with coefficients in L, and let x ∈ L. By Theorem 1 there are a_{1}, …, a_{m} ∈ K such that v_{t}(x − f(a_{1}, …, a_{m})) is maximal. Since v_{t}L = v_{t}𝔽_{q}((t)) we can choose c ∈ L such that v_{t}c = v_{t}(x − f(a_{1}, …, a_{m})). So the existential sentence ∃x_{1} … ∃x_{m} v_{t}c = v_{t}(x − f(x_{1}, …, x_{m})) holds in (𝔽_{q}((t)), v_{t}). But then it also holds in (L, v_{t}|_{L}), that is, a_{1}, …, a_{m} can be chosen to lie in L.

An example of such a valued subfield is the henselization of 𝔽_{q}(t) inside 𝔽_{q}((t)), where both fields carry the t-adic valuation.

Remark 2. Theorem 1 does not hold for 𝔽_{q}(t) with its t-adic valuation: the additive polynomial X − X^{p} does not have t in its image on this field, but t can be approximated arbitrarily closely by elements in this image, since t = x − x^{p} for x = ∑_{n=0}^{∞} t^{p^{n}} ∈ 𝔽_{q}((t)).

Remark 3. The proofs of Lemmas 3 and 4 do not use the local compactness of (K, v) = (𝔽_{q}((t)), v_{t}). Actually, we just need that 1, t, …, t^{p−1} is a basis for K|K^{p} and that vt is not divisible by p in vK. Hence, in every extension (L, v) of (𝔽_{q}((t)), v_{t}) with these properties one can derive polynomials g_{1}, …, g_{m} over L as in Lemma 4 from any given additive polynomial f over L. The proof of Lemma 5 then shows that there exists α_{0} ∈ vL such that for each α ≤ α_{0} in vL, any complement C of B_{α}(0) in L, and all a_{1}, …, a_{m} ∈ C we have

v ∑_{i=1}^{m} g_{i}(a_{i}) = min_{i} vg_{i}(a_{i})

(so the sum g_{1}(C) + … + g_{m}(C) is “valuation direct”). If (L, v) is maximal, one can then prove along the lines of [K] that g_{1}(C) + … + g_{m}(C) has OA in (L, v).

References

[G] M. J. Greenberg, Rational points in henselian discrete valuation rings. Publ. Math. I.H.E.S. 31 (1966), 59–64

[K] F.-V. Kuhlmann, Elementary properties of power series fields over finite fields. To appear in J. Symb. Logic

[L] S. Lang, Algebra. Addison-Wesley, New York (1965)

Department of Mathematics, University of Illinois at Urbana, 273 Altgeld Hall, 1409 West Green Street, Urbana, IL 61801, USA email: vddries@math.uiuc.edu

Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6 email: fvk@math.usask.ca