Simple Transcendental Extension Research Articles

Key polynomials for simple extensions of valued fields

Let $\iota:K\hookrightarrow L\cong K(x)$ be a simple transcendental extension of valued fields, where $K$ is equipped with a valuation $\nu$ of rank 1. That is, we assume given a rank 1 valuation $\nu$ of $K$ and its extension $\nu'$ to $L$. Let $(R_\nu,M_\nu,k_\nu)$ denote the valuation ring of $\nu$. The purpose of this paper is to present a refined version of MacLane's theory of key polynomials, similar to those considered by M. Vaqui\'e, and reminiscent of related objects studied by Abhyankar and Moh (approximate roots) and T.C. Kuo. Namely, we associate to $\iota$ a countable well ordered set $$ \mathbf{Q}=\{Q_i\}_{i\in\Lambda}\subset K[x]; $$ the $Q_i$ are called {\bf key polynomials}. Key polynomials $Q_i$ which have no immediate predecessor are called {\bf limit key polynomials}. Let $\beta_i=\nu'(Q_i)$. We give an explicit description of the limit key polynomials (which may be viewed as a generalization of the Artin--Schreier polynomials). We also give an upper bound on the order type of the set of key polynomials. Namely, we show that if $\operatorname{char}\ k_\nu=0$ then the set of key polynomials has order type at most $\omega$, while in the case $\operatorname{char}\ k_\nu=p>0$ this order type is bounded above by $\omega\times\omega$, where $\omega$ stands for the first infinite ordinal.

ON LIFTINGS OF POWERS OF IRREDUCIBLE POLYNOMIALS

Let v be a henselian valuation of arbitrary rank of a field K with valuation ring Rv having maximal ideal Mv. Using the canonical homomorphism from Rv onto Rv/Mv, one can lift any monic irreducible polynomial with coefficients in Rv/Mv to yield monic irreducible polynomials over Rv. Popescu and Zaharescu extended this approach and introduced the notion of lifting with respect to a residually transcendental prolongation w of v to a simple transcendental extension K(x) of K. As it is well known, the residue field of such a prolongation w is [Formula: see text], where [Formula: see text] is the residue field of the unique prolongation of v to a finite simple extension L of K and Y is transcendental over [Formula: see text] (see [V. Alexandru, N. Popescu and A. Zaharescu, A theorem of characterization of residual transcendental extension of a valuation, J. Math. Kyoto Univ.28 (1988) 579–592]). It is known that a lifting of an irreducible polynomial belonging to [Formula: see text] with respect to w, is irreducible over K. In this paper, we give some sufficient conditions to ensure that a given polynomial in K[x] satisfying these conditions which is a lifting of a power of some irreducible polynomial belonging to [Formula: see text] with respect to w, is irreducible over K. Our results extend Eisenstein–Dumas and generalized Schönemann irreducibility criteria.

ON IRREDUCIBLE FACTORS OF POLYNOMIALS OVER COMPLETE FIELDS

Let (K, v) be a complete rank-1 valued field. In this paper, we extend classical Hensel's Lemma to residually transcendental prolongations of v to a simple transcendental extension K(x) and apply it to prove a generalization of Dedekind's theorem regarding splitting of primes in algebraic number fields. We also deduce an irreducibility criterion for polynomials over rank-1 valued fields which extends already known generalizations of Schönemann Irreducibility Criterion for such fields. A refinement of Generalized Akira criterion proved in Khanduja and Khassa [Manuscripta Math.134(1–2) (2010) 215–224] is also obtained as a corollary of the main result.

On prolongations of valuations via Newton polygons and liftings of polynomials

Let v be a real valuation of a field K with valuation ring Rv. Let K(θ) be a finite separable extension of K with θ integral over Rv and F(x) be the minimal polynomial of θ over K. Using Newton polygons and residually transcendental prolongations of v to a simple transcendental extension K(x) of K together with liftings with respect to such prolongations, we describe a method to determine all prolongations of v to K(θ) along with their residual degrees and ramification indices over v. The problem is classical but our approach uses new ideas. The paper gives an analogue of Ore’s Theorem when the base field is an arbitrary rank-1 valued field and extends the main result of [S.D. Cohen, A. Movahhedi, A. Salinier, Factorization over local fields and the irreducibility of generalized difference polynomials, Mathematika 47 (2000) 173–196].

Note on a Variation of the Schröder-Bernstein Problem for Fields

In this note we study fields F with the property that the simple transcendental extension F(u) of F is isomorphic to some subfield of F but not isomorphic to F. Such a field provides one type of solution of the Schröder-Bernstein problem for fields.

On minimal pairs and residually transcendental extensions of valuations

In this paper, further insight is obtained into the earlier approach of studying residually transcendental extensions of a valuation v of a field K to a simple transcendental extension K(x) of K by means of minimal pairs, thereby introducing new invariants corresponding to any element of an algebraic closure of K. It is also shown that these invariants are of independent interest as well. A characterization of those elements a belonging to is given such that there exists a minimal pair (a, δ) for some δ in the divisible closure of the value group of v.

On Solutions of Linear Ordinary Difference Equations in their Coefficient Field

We extend the notion of monomial extensions of differential fields, i.e. simple transcendental extensions in which the polynomials are closed under differentiation, to difference fields. The structure of such extensions provides an algebraic framework for solving generalized linear difference equations with coefficients in such fields. We then describe algorithms for finding the denominator of any solution of those equations in an important subclass of monomial extensions that includes transcendental indefinite sums and products. This reduces the general problem of finding the solutions of such equations in their coefficient fields to bounding their degrees. In the base case, this yields in particular a new algorithm for computing the rational solutions of q -difference equations with polynomial coefficients.

A Uniqueness Problem in Valued function Fields of Conics

Let v0 be a valuation of a field K0 with value group G0. Let K be a function field of a conic over K0, and let v be an extension of v0 to K with value group G such that G/G0 is not a torsion group. Suppose that either (K0, v0) is henselian or v0 is of rank 1, the algebraic closure of K0 in K is a purely inseparable extension of K0, and G0 is a cofinal subset of G. In this paper, it is proved that there exists an explicitly constructible element t in K, with v(t) non-torsion modulo G0 such that the valuation of K0(t), obtained by restricting v, has a unique extension to K. This generalizes the result proved by Khanduja in the particular case, when K is a simple transcendental extension of K0 (compare [4]). The above result is an analogue of a result of Polzin proved for residually transcendental extensions [8].

A uniqueness problem in simple transcendental extensions of valued fields

Let υ0 be a valuation of a field K0 with value group G0 and υ be an extension of υ0 to a simple transcendental extension K0(x) having value group G such that G/G0 is not a torsion group. In this paper we investigate whether there exists t∈K0(x)/K0 with υ(t) non-torsion mod G0 such that υ is the unique extension to K0(x) of its restriction to the subfield K0(t). It is proved that the answer to this question is “yes” if υ0 is henselian or if υ0 is of rank 1 with G0 a cofinal subset of the value group of υ in the latter case, and that it is “no” in general. It is also shown that the affirmative answer to this problem is equivalent to a fundamental equality which relates some important numerical invariants of the extension (K, υ)/(K0, υ0).

On valuations of K(x)

For a valued field (K, v), let Kv denote the residue field of v and Gv its value group. One way of extending a valuation v defined on a field K to a simple transcendental extension K(x) is to choose any α in K and any μ in a totally ordered Abelian group containing Gv, and define a valuation w on K[x] by w(Σici(x – α)i) = mini (v(ci) + iμ). Clearly either Gv is a subgroup of finite index in Gw = Gv + ℤμ or Gw/Gv is not a torsion group. It can be easily shown that K(x)w is a simple transcendental extension of Kv in the former case. Conversely it is well known that for an algebraically closed field K with a valuation v, if w is an extension of v to K(x) such that either K(x)w is not algebraic over Kv or Gw/Gv is not a torsion group, then w is of the type described above. The present paper deals with the converse problem for any field K. It determines explicitly all such valuations w together with their residue fields and value groups.

Prolongations of valuations to simple transcendental extensions with given residue field and value group

Let K 0 (x) be a simple transcendental extension of a field K 0 , υ 0 be a valuation of K 0 with value group G 0 and residue field K 0 . Suppose G 0 ⊆G 1 ⊆G is an inclusion of totally ordered abelian groups with [G 1 : G 0 ] 1 and an infinite cyclic group. It is proved that there exists an (explicitly constructible) valuation υ of K 0 (x) extending υ 0 such that the value group of υ is G and its residue field is k, where k is a given finite extension of k 0 . This is analogous to a result of Matignon and Ohm [2, Corollary 3.2] for residually non-algebraic prolongations of υ 0 to K 0 (x).

Value groups and simple transcendental extensions

Introduction. Throughout the paper K(x) is a simple transcendental extension of a field K; v is a valuation of K and w is an extension of v to K(x). Also koIk and GoIG denote respectively the residue fields and the value groups of the valuations v and w. A well-known theorem conjectured by Nagata asserts that either k is an algebraic extension of feo or k is a simple transcendental extension of a finite extension of ko (cf [4] or [6] or [1, Corollary 2.3]). We prove here an analogous result for the value groups viz. either G/ Go is a torsion group or there exists a subgroup G1 of G containing Go with [G1: Go] > ∞ such that G is the direct sum of G1 and an infinite cyclic group. Incidentally we obtain a description of the valuation w as well as of its residue field in the second case. Thus a characterization of all those extensions w of v to K(x), for which w(K(x)\{0})/Go is not a torsion group, is given. Corresponding to such a valuation w, we define three numbers N, S and T which satisfy the inequality N ≥= ST. This is analogous to the fundamental inequality established by Ohm (cf. [5, 1.2]) for residually transcendental extensions of v to K(x). We also investigate the conditions under which N = ST

On extensions of valuations to simple transcendental extensions

Let ν0 be a valuation of a field K0 with residue field k0 and value group Z, the group of rational integers. Let K0(x) be a simple transcendental extension of K0. In 1936, Maclane [3] gave a method to determine all real valuations V of K0(x) which are extensions of ν0. But his method does not seem to give an explicit construction of these valuations. In the present paper, assuming K0 to be a complete field with respect to ν0, we explicitly determine all extensions of ν0 to K0(x) which have Z as the value group and a simple transcendental extension of k0 as the residue field. If V is any extension of ν0 to K0(x) having Z as the value group and a transcendental extension of k0 as the residue field, then using the Ruled Residue theorem [4, 2, 5], we give a method which explicitly determines V on a subfield of K0(x) properly containing K0.

ON THE EXTENSION OF THE RING TOPOLOGY OF A $ \sigma$-BOUNDED FIELD TO A SIMPLE TRANSCENDENTAL EXTENSION OF THE FIELD

If is a ring topology of a field such that is the union of countably many bounded sets, then there exists a ring topology on a simple transcendental extension of such that is the union of countably many bounded sets and is the restriction of to .Bibliography: 6 titles.

A structure theorem for simple transcendental extensions of valued fields

The fundamental inequality for a finite algebraic extension of a valued field relates the degree of the extension to the ramification indices and residue degrees, and of primary importance is the question of when this inequality becomes equality. An analogous question for simple transcendental extensions is treated here as an application of a fundamental structure theorem for such extensions.

The Ruled Residue Theorem for Simple Transcendental Extensions of Valued Fields

The Ruled Residue Theorem for Simple Transcendental Extensions of Valued Fields

The ruled residue theorem for simple transcendental extensions of valued fields

A proof is given for the Ruled Residue Conjecture, which asserts that if υ \upsilon is a valuation of a simple transcendental field extension K 0 ( x ) {K_0}(x) and υ 0 {\upsilon _0} is the restriction of υ \upsilon to K 0 {K_0} , then the residue field of υ \upsilon is either ruled or algebraic over the residue field of υ 0 {\upsilon _0} .

Extending locally bounded field topologies to a simple transcendental extension

Extending locally bounded field topologies to a simple transcendental extension

Transcendental extensions of field topologies on countable fields

Let be a field topology on a countable field K and let K(x) be a simple transcendental extension of K. Then there exists a field topology 9' for K(x) such that 'l K=Y. Let K be a countable field and let 91 be a fundamental system of neighborhoods at zero. We identify K with the field of constant functions on K and K(x) with the field R(K) of rational functions over K. If D is a nonprincipal filter on K, then we define for each U E X, UD to be the set of all f E R(K) with {rjf(r) E U} E D and 91D to be the set {UDjU E %}. 1D1K=9, and % D is a filter base. But 91D is not necessarily a fundamental system of neighborhoods at zero of a field topology on R(K). 1. DEFINITION. A filter D on K is called 91-generic, if for each U E 91 and for eachf E R(K) there is a V E W such that {rIf(r)Vc U} E D. 2. THEOREM. If D is a 9-genericfilter on K, then %D is a fundamental system on R(K) such that 91D1K=9. PROOF. To see that %D defines a group topology on R(K), let UD C SD be given. Since 91 is a fundamental system there is a V such that VVc U. Suppose f, g E VD. Then {rjf(r) E V} E D and {r|g(r) E V} E D. By {rlf(r)-g(r) e U}{rlf(r) E V} n{r|g(r) E V} we have that f-g E UD . Thus VD_ VDC UD. By a similar argument, it can be seen that inversion and multiplication at zero are continuous. So it remains to show that multiplication is continuous everywhere. Let f cR(K) and UD C91D be given. Since D is 9-generic there is a V such that {rlf(r) Vc U} E D. Suppose g E VD. Then {r|g(r) e V} e D. {rjg(r) f (r) c( U}) {rJf (r) * Vc U})r{rjg(r) e V} and therefore f -g E UD. Thus, multiplication is continuous everywhere. O Received by the editors March 2, 1972 and, in revised form, July 26, 1972. AMS (MOS) subject classifications. Primary 12J99. ? American Mathematical Society 1973

A Theorem on Valuation Rings and its Applications

In the present paper, we first prove the following THEOREM 1. Let K be a field, x a transcendental element over K, and V* a valuation ring of K(x). Set V= V* ∩ K. Denote by p* and p the maximal ideals of V* and V respectively. If (i) V*/p* is not algebraic over V/p and (ii) the value group of V* is isomorphic to Zn (Z=the module of rational integers), i.e., V* is of rank n and discrete in the generalized sense, then V*/p* is a simple transcendental extension of a finite algebraic extension of V/p.