Selçuk J. Appl. Math. Selçuk Journal of Vol. 12. No. 2. pp. 111-117, 2011 Applied Mathematics
On Residual Transcendental Extensions of a Valuation withrankv = 2 Burcu Ozturk, Figen Oke
Trakya University Department of Mathematics, 22030 Edirne, Turkiye e-mail: burcinburcu2002@ yaho o.com,figenoke@ gm ail.com
Received Date:August 12, 2011 Accepted Date: December 15, 2011
Abstract. Let v = v1◦ v2 be a valuation of a field K with rankv = 2. In this paper a residual transcendental extension w = w1◦ w2 of v to K(x) is studied where w1 and w2 are the residual extensions of v1 and v2 respectively. A characterization of lifting polynomials is given and the constants w(K,v)(c), ∆(K,v)(c), δ(K,v)(c) are defined for c, where (c, δ) ∈ K × Gv is the minimal pair defining w.
Key words: Valued Fields, Residual Transcendental Extensions, Lifting Poly-nomials, Krasner’s constant.
2000 Mathematics Subject Classification: 12F05, 12J10, 12J20. 1. Introduction
Let (K, v1) be a henselian valued field, v2 be a valuation of residue field kv1
and v = v1◦ v2 be composite of valuations v1 and v2. Let w1 be a residual transcendental extension of v1to K(x) defined by minimal pair (a1, δ1), w2 be a residual transcendental extension of v2 to the residue field kw1 defined by
minimal pair (a2, δ2). Let f (x) = Irr(a1, K) be the minimal polynomial of a1 with respect to K and g(Y ) = Irr(a2, kv1) be the minimal polynomial of a2with
respect to kv1. Let G(x) ∈ K[x] be a lifting of the polynomial g(Y ) with respect
to w1 and c be a root of G(x). It is known that (c, a1) is a distinguished pair with respect to v1. Distinguished pairs, distinguished chains and their relations with lifting polynomials firstly studied by Popescu and Zaharescu in 1995 [3]. The constants of an algebraic element are important for defining extensions of v to rational function field K(x), giving characterizations of tame extensions and for using in the definition of distinguished pairs. In this paper, the definition of the residual transcendental extension w of v with rankv = 2 to K(x) is studied by using a root of a lifting polynomial and the Krasner’s constant of c is obtained as w(K,v)(c) = (w(K,v1)(c), w(kv1,v2)(a2)) and the other constants
of c are obtained as ∆(K,v)(c) = (∆(K,v1)(c), ∆(kv1,v2)(a2)) and δ(K,v)(c) =
(δ(K,v1)(c), δ(kv1,v2)(a2)). Also a relation on the lifting polynomials with respect
to w with rankw = 2 is given by using the lifting polynomials with respect to w1 and w2 with rankw1= rankw2= 1.
2. Notations and Some Preliminaries
Throughout this paper, v is a henselian valuation of K with value group Gv, residue field kv, valuation ring Ov and ¯v is the fixed extension of v to the algebraic clousure K of K. For any d in the valuation ring of v, d∗
v will denote its residue, i.e. the image of d under the canonical homomorphism from the valuation ring of v onto the residue field kv. If d isn’t an element of valuation ring then for an element β ∈ K∗ such that v(d) = v(β), d∗
v will denote the v−residue of d/β. If L is a finite extension of K, w is an extension of v to L then e(w/v) = [Gw: Gv] is the ramification index and f (w/v) = [kw: kv] is the residue degree of w/v. In this paper we shall use the notations given in [1] and [2] in general.
Let K(x) be a field of rational functions of one variable over K. The extension w of v to K(x) is called a residual transcendental (r.t.) extension of v if kwis a transcendental extension of kv. If w is a residual transcendental extension of v to K(x) then w is defined by minimal pair (a, δ) ∈ K ×Gvwhere a is seperable over K. An element (a, δ) ∈ K × Gv is called minimal pair with respect to (K, v) if every b ∈ K, v(a − b) ≥ δ implies [K(a) : K] ≤ [K(b) : K]. Let f(x) = Irr(a, K) be minimal polynomial of a over K and γ =P
a0 inf(δ, v(a − a
0)) where a0 runs over all roots of f (x). Each polynomial F (x) ∈ K[x] can be written uniquelly as F (x) =P
i
Fi(x)f (x)i, deg Fi< deg f and the residual transcendental extension w of v to K(x) is defined as;
w(F ) = infi(v(Fi(a)) + iγ)
Let e be the smallest positive integer such that eγ ∈ Gva and h(x) ∈ K [x] such
that degh < deg f , v(h(a)) = eγ where va is the restiriciton of ¯v to K(a). Let r = fe/h ∈ K(x) such that w(r) = 0. So r∗ is transcendental over the residue field kv and also kw= kva(r∗) is the residue field of w. We will denote r∗ by Y .
Let g(Y ) ∈ kva[Y ]. G(x) ∈ K [x] is a lifting of the polynomial g(Y ) with respect
to w if the following conditions are satisfied; 1. deg G = e. deg g. deg f
2. w(G) = e. deg g.γ 3. ¡ G
hdeg g
¢∗ w= g.
Krasner’s constant of an element α ∈ K\K is defined by
and the other constants are defined as; ∆(K,v)(α) = min {v(α − α0) |α0is a K−conjugate of α } , δ(K,v)(α) = sup © v (α − β)¯¯ β ∈ K, [K (β) : K] < [K (α) : K]ª. 3. Main Results
Let (K, v1) be a henselian valued field, v2 be a valuation of residue field kv1
and v = v1◦ v2 be composite of valuations v1 and v2. Let w1 be a residual transcendental extension of v1 to the rational function field K(x) defined by minimal pair (a1, δ1), w2 be a residual transcendental extension of v2 to the residue field kw1defined by minimal pair (a2, δ2) and w = w1◦w2be an extension
of v to K(x) which is composite of valuations w1and w2. Let f (x) = Irr(a1, K) be a minimal polynomial of a1 with respect to K, w1(f ) = γ1 and e1 be the smallest positive integer such that e1γ1 ∈ Gva1 where va1 is the restriction of
v1 to K(a1) and h1(x) ∈ K [x] such that deg h1 < deg f , va1(h1(a1)) = e1γ1. Let g(Y ) = Irr(a2, kv1) be a minimal polynomial of a2 with respect to kv1,
w2(g) = γ2 and e2 is the smallest positive integer such that e2γ2∈ Gva2 where va2 is the restriction of v2 to the field kv1(a2) and h2(x) ∈ kva1[Y ] such that
deg h2< deg g, va2(h2(a2)) = e2γ2 where w1− residue of fe1/h1 is Y and w2− residue of ge2/h
2 is Z.
Theorem 3.1. [2] Let G(x) be a lifting polynomial of g(Y ) with respect to w1 and λ = (δ1, δ2). Under the above notations there exists a root c of the polynomial G in K such that (c, λ) is a minimal pair with respect to (K, v) and w is defined by (c, λ) and v.
Under the above notations we have the following theorems. The first theorem is obtained as a natural result of Theorem 3.1.
Theorem 3.2. Let Irr(a2, kv1) = g(Y ) ∈ kva1[Y ] and G(x) ∈ K[x] be a lifting
of the polynomial g(Y ) with respect to valuation w1 and c be a root of G(x) that defines w with λ and v . If F (x) ∈ K[x] and F (x) =P
i
Fi(x)G(x)i, is the G−expansion of F then the valuation w is defined as;
w(F (x)) = infi à v1(Fi(a1)) + i.e1. deg g.γ1, w2 õ Fi(x) Fi(a) ¶∗ w1 ! + i.γ2 ! .
Proof. The value group and the residue field of w are Gw = Gw1 × Gw2
(ordered lexicographically) and kw= kw2 = kva2(Z) respectively where Gw1 =
Gva1 + Z.γ1 and Gw2 = Gva2+ Z.γ2. Since G(x) is a lifting of the polynomial
1. deg G = e1. deg g. deg f 2. w1(G) = e1. deg g.γ1 3. ³ G (h1)deg g ´∗ w1 = g
We suppose that g(Y ) is a lifting of a polynomial H(Z) ∈ kva2[Z] with
re-spect to w2. Thus we have deg g = e2. deg H. deg g, w2(g) = e2. deg H.γ2, ³
g (h2)deg H
´∗ w2
= H. Then e2= 1 and also the image of the polynomial G(x) ∈ K[x] under the valuation w is
w(G) = (w1(G), w2(g)) = (e1. deg g.γ1, γ2)
Let e be the smallest positive integer such that e.w(G) ∈ Gvc where vc is the
restriction of ¯v to K(c). Since Gvc ⊇ Gva1 × Gva2, e1γ1∈ Gva1 and γ2∈ Gva2
then w(G) = (e1. deg g.γ1, γ2) ∈ Gvc i.e. e = 1. Each F ∈ K [x] is written
uniquelly as F (x) =P i
Fi(x)G(x)i, Fi(x) ∈ K[x], deg Fi(x) < deg G and we have
w(F (x)) = infi{w(Fi(x)) + i.w(G(x))} Since w1(Fi(x)) = w1(Fi(a1)) from [4] then
³F i(x) Fi(a1) ´∗ w1 6= 0. Therefore w(F (x)) = infi (à w1(Fi(x)), w2 õ Fi(x) Fi(a1) ¶∗ w1 !! + i. (e1. deg g.γ1, γ2) ) = infi (à v1(Fi(a1)) + i.e1. deg g.γ1, w2 õ Fi(x) Fi(a1) ¶∗ w1 ! + i. γ2 !)
is obtained as desired. Also from [5] , using by the equality v1(Fi(a1)) = v1(Fi(c)) it can be written as w(F (x)) = infi (à v1(Fi(c)) + i.e1. deg g.γ1, w2 õ Fi(x) Fi(c) ¶∗ w1 ! + i. γ2 !)
Now we have three residual transcendental extensions. w1 is residual tran-scendental extension of v1, w2 is residual transcendental extension of v2 and w = w1◦ w2 is residual transcendental extension of v = v1◦ v2. Then we can define lifting polynomials for each residual transcendental extension. In the following theorem, the relation between these lifting polynomials is given. Theorem 3.3. Let J(Z) ∈ kva2[Z], T (Y ) ∈ kva1[Y ] be a lifting polynomial of J(Z) with respect to valuation w2 and P (x) ∈ K[x] be a lifting polynomial
of T (Y ) with respect to w1. Then P (x) is a lifting polynomial of J(Z) with respect to w.
Proof. The following equations are obtained using the definition of lifting deg T = e2. deg g. deg J, w2(T ) = e2. deg J. γ2
(1) Ã T (h2)deg J !∗ w2 = J
deg P = e1. deg f. deg T, w1(P ) = e1. deg T. γ1
(2) Ã P (h1)deg T !∗ w1 = T
From Theorem 3.2. it is known that e = e2= 1 . So it is easily seen that deg P = e1. deg f. deg g. deg J = deg G. deg J = e. deg G. deg J and then
w(P ) = (w1(P ), w2(T )) = (e1. deg T. γ1, deg J. γ2)
= (e1. deg g. deg J. γ1, deg J. γ2) = e. deg J. (w1(G), w2(g)) = e. deg J.w(G) is obtained. There is a polynomial h(x) ∈ K [x], deg h < deg G such that w(G) = v(h(c)) [3]. Then one has
w(h) = (e1. deg g. γ1, γ2) = ³ w1(h1deg g), w2(h2) ´ Therefore (3) w1(h) = w1(h1deg g)and µ h h1deg g ¶∗ w1 = h2
are obtained. Keeping in view of (1) and (2) we have the equality
J = ⎛ ⎜ ⎝ ³ P h1deg T ´∗ w1 h2deg J ⎞ ⎟ ⎠ ∗ w2 Thus from (3)
J = ⎛ ⎜ ⎜ ⎜ ⎝ ³ P (h1deg g)deg J ´∗ w1 µ³ h h1deg g ´deg J¶∗ w1 ⎞ ⎟ ⎟ ⎟ ⎠ ∗ w2 = µ P hdeg J ¶∗ w
is obtained. Hence P (x) is a lifting of the polynomial J(Z) with respect to w. In the following theorem the constants ∆(K,v)(α), w(K,v)(α), δ(K,v)(α) are ob-tained for the element c ∈ K, where (c, λ) ∈ K × Gv is the minimal pair of definition of w. Remember that these constants are used for defining the resid-ual transcendental extension of any valuation.
Theorem 3.4. Let Irr(a2, kv1) = g(Y ) ∈ kva1[Y ] and G(x) ∈ K [x] be a lifting
of the polynomial g(Y ) with respect to the valuation w1and c be a root of G(x). Let v1 be the fixed extension of v1 to the algebraic clousure K of K, v2 be the fixed extension of v2 to the algebraic clousure kv1 of kv1 and v = v1◦ v2 be the
fixed extension of v to K which is a composite of valuations v1 and v2. Then the costants of c ∈ K with respect to v are obtained as;
∆(K,v)(c) = (∆(K,v1)(c), ∆(kv1,v2)(a2))
w(K,v)(c) = (w(K,v1)(c), w(kv1,v2)(a2))
δ(K,v)(c) = (δ(K,v1)(c), δ(kv1,v2)(a2))
Proof. Since rankw = 2, then the rank of the value group Gwis 2 and so Gw is lexicografically ordered. Keeping in view of the polynomial G(x) = Irr(c, K) is a lifting of the polynomial g(Y ) = Irr(a2, kv1), the folowing equations are
obtained by [2] and using Hensel’s Lemma.
∆(K,v)(c) = min {v(c − c0) |c0 is a K−conjugate of c } = min©(v1(c − c0), v2((c − c0)∗v1)) |c 0 is a K−conjugate of cª = min {(v1(c − c0), v2(a2− a02)) |c0 is a K−conjugate of c, a02 is a kv−conjugate of a2} = (min {v1(c − c0) | c0is a K−conjugate of c } , min {v2(a2− a02) | a02 is a kv−conjugate of a2}) = (∆(K,v1)(c), ∆(kv1,v2)(a2))
Krasner’s constant of c is below: w(K,v)(c) = max {v(c − c0) |c0 is a K−conjugate of c , c06= c} = max©(v1(c − c0), v2((c − c0)∗v1)) |c 0 is a K−conjugate of c, c06= cª = max {(v1(c − c0), v2(a2− a02)) |c0 is a K−conjugate of c, c06= c, a0 2 is a kv−conjugate of a2, a026= a2} = (max {v1(c − c0) | c0is a K−conjugate of c, c0 6= c } , max {v2(a2− a02) | a02is a kv−conjugate of a2, a026= a2}) = (w(K,v1)(c), w(kv1,v2)(a2))
Finally we have that; δ(K,v)(c) = sup © v(c − β)¯¯ β ∈ K , [K(β) : K] < [K(c) : K]ª = sup©(v1(c − β), v2((c − β)∗v1) ¯ ¯ β ∈ K , [K(β) : K] < [K(c) : K]ª = sup©(v1(c − β), v2(a2− β∗v1) ¯ ¯ β ∈ K , [K(β) : K] < [K(c) : K]ª =¡sup©v1(c − β) ¯ ¯ β ∈ K , [K(β) : K] < [K(c) : K]ª , sup©(v2(a2− β∗v1) ¯ ¯ β∗ v1 ∈ K , £ kv1(β∗v1) : kv1 ¤ < [kv1(a2) : kv1] ª ¢ = (δ(K,v1)(c), δ(kv1,v2)(a2)) References
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