ON EISENSTEIN-DUMAS AND GENERALIZED SCHÖNEMANN POLYNOMIALS

Transcription

1 ON EISENSTEIN-DUMAS AND GENERALIZED SCHÖNEMANN POLYNOMIALS Anuj Bshno and Sudsh K. Khanduja Dpartmnt of Mathmatcs, Panjab Unvrsty, Chandgarh , Inda. E-mal: ABSTRACT. Lt v b a valuaton of a fld K havng valu group Z. It s known that a polynomal x n + a n 1 x n a 0 satsfyng v(a ) v(a 0) > 0 n n wth v(a 0 ) coprm to n, s rrducbl ovr K. Such a polynomal s rfrrd to as an Esnstn-Dumas polynomal wth rspct to v. In ths papr, w gv ncssary and suffcnt condtons so that som translat g(x+a) of a gvn polynomal g(x) blongng to K[x] s an Esnstn-Dumas polynomal wth rspct to v. In fact an analogous problm s dalt wth for a wdr class of polynomals, vz. Gnralzd Schönmann polynomals wth coffcnts ovr valud flds of arbtrary rank. Kywords : Fld Thory and polynomals; Valud flds; Non-Archmdan valud flds Mathmatcs Subjct Classfcaton : 12E05; 12J10; 12J25. All corrspondnc may b addrssd to ths author. 1

2 1. INTRODUCTION Th classcal Schönmann Irrducblty Crtron stats that f f(x) s a monc polynomal wth coffcnts from th rng Z of ntgrs whch s rrducbl modulo a prm numbr p and f g(x) blongng to Z[x] s a polynomal of th form g(x) = f(x) + pm(x) whr M(x) blongng to Z[x] s rlatvly prm to f(x) modulo p and th dgr of M(x) s lss than th dgr of g(x), thn g(x) s rrducbl ovr th fld Q of ratonal numbrs. Such a polynomal s rfrrd to as a Schönmann polynomal wth rspct to p and f(x). It can b asly sn that f g(x) s as abov, thn th f(x)-xpanson of g(x) obtand on dvdng t by succssv powrs of f(x) gvn by g(x) = g (x)f(x), dg g (x) < dgf(x), satsfs () g (x) = 1, () p dvds th contnt of ach polynomal g (x) for 0 1 and () p 2 dos not dvd th contnt of g 0 (x). Clarly any polynomal g(x) blongng to Z[x] whos f(x)-xpanson satsfs th abov thr proprts s a Schönmann polynomal wth rspct to p and f(x). Not that a monc polynomal s an Esnstn polynomal wth rspct to a prm p f and only f t s a Schönmann polynomal wth rspct to p and f(x) = x. Th Schönmann Irrducblty Crtron has bn xtndd to polynomals wth coffcnts ovr valud flds n svral ways (cf. Khanduja and Saha, 1997; Rbnbom, 1999, Chaptr 4, D; Brown, 2008). In 2008, Ron Brown gav a gnralzaton of th Schönmann Irrducblty Crtron for polynomals wth coffcnts n a valud fld (K, v) of arbtrary rank, whch wll b statd aftr ntroducng som notatons. W shall dnot by v x th Gaussan valuaton of th fld K(x) of ratonal functons n an ndtrmnat x whch xtnds th valuaton v of K and s dfnd on K[x] by v x ( a x ) = mn{v(a ) a K}. For an lmnt ξ n th valuaton rng R v of v wth maxmal dal M v, ξ wll dnot ts v-rsdu,.., th mag of ξ undr th canoncal homomorphsm from R v onto R v /M v. For f(x) blongng to R v [x], f(x) wll stand for th polynomal ovr R v /M v obtand by rplacng ach coffcnt of f(x) by ts v-rsdu. Th followng rsult of Ron Brown whch gnralzs th Schönmann Irrducblty Crtron s provd n scton 3 (s Lmma 3.1). 2

3 Thorm A. Lt v b a valuaton of arbtrary rank of a fld K wth valu group G and valuaton rng R v havng maxmal dal M v. Lt f(x) blongng to R v [x] b a monc polynomal of dgr m such that f(x) s rrducbl ovr R v /M v. Assum that g(x) R v [x] s a monc polynomal whos f(x)-xpanson 1 f(x) + g (x)f(x) satsfs vx (g (x)) vx (g 0 (x)) > 0 for 0 1 and v x (g 0 (x)) / dg for any numbr d > 1 dvdng. Thn g(x) s rrducbl ovr K. A polynomal satsfyng th hypothss of Thorm A wll b rfrrd to as a Gnralzd Schönmann polynomal wth rspct to v and f(x). In th partcular cas whn f(x) = x, t wll b calld an Esnstn-Dumas polynomal wth rspct to v. Whn v s a dscrt valuaton wth valu group Z, thn a monc polynomal a x s an Esnstn-Dumas polynomal wth rspct to v f v(a ) v(a 0) and v(a 0 ) s coprm to. Thus Thorm A xtnds th usual Esnstn-Dumas Irrducblty Crtron 1. In ths papr, w frst nvstgat whn a translat g(x+a) of a gvn polynomal g(x) blongng to K[x] havng a root θ s an Esnstn-Dumas polynomal wth rspct to an arbtrary hnslan valuaton v of a fld K. It s shown that g(x + a) s such a polynomal f and only f K(θ)/K s a totally ramfd xtnson and (θ, a) s a (K, v)-dstngushd par as dfnd blow. In partcular, t s dducd that f som translat of a polynomal g(x) = x + a 1 x a 0 s an Esnstn-Dumas polynomal wth rspct to v wth not dvsbl by th charactrstc of th rsdu fld of v, thn th polynomal g(x a 1 ) s an Esnstn-Dumas polynomal wth rspct to v. Ths gnralzs a rsult of M. Juras provd n 2006 (cf. Juras, 2006). W also dal wth th followng mor gnral problm rlatd to Thorm A. Lt g(x) blongng to R v [x] b a monc polynomal ovr a hnslan valud fld (K, v) of arbtrary rank wth ḡ(x) = φ(x) whr φ(x) s an rrducbl polynomal ovr R v /M v and θ s a root of g(x). What ar ncssary and suffcnt condtons so that g(x) s a Gnralzd Schönmann polynomal wth rspct to v and som polynomal f(x) R v [x] wth f(x) = φ(x)? 1 Esnstn-Dumas Irrducblty Crtron. Lt g(x) = a n x n + a n 1 x n a o b a polynomal wth coffcnts n Z. Suppos thr xsts a prm p whos xact powr p r dvdng a (whr r = f a = 0), satsfy r n = 0, (r /n ) (r 0 /n) for 0 n 1 and r 0, n ar coprm. Thn g(x) s rrducbl ovr Q. 3

4 Our rsults ar provd usng saturatd dstngushd chans whch wll b dfnd aftr ntroducng som notatons. In what follows, v s a hnslan valuaton of arbtrary rank of a fld K and ṽ s th unqu prolongaton of v to th algbrac closur K of K wth valu group G. By th dgr of an lmnt α n K, w shall man th dgr of th xtnson K(α)/K and shall dnot t by dg α. For an lmnt ξ n th valuaton rng of ṽ, ξ wll stand for ts ṽ-rsdu and for a subfld L of K, L and G(L) wll dnot rspctvly th rsdu fld and th valu group of th valuaton of L obtand by rstrctng ṽ. Whn thr s no chanc of confuson, w shall wrt ṽ(α) as v(α) for α blongng to K. A fnt xtnson (K, v )/(K, v) s calld dfctlss f [K : K] = f, whr and f ar th ndx of ramfcaton and th rsdual dgr of v /v. Rcall that a par (θ, α) of lmnts of K s calld a dstngushd par (mor prcsly (K, v)-dstngushd par) f th followng thr condtons ar satsfd: () dg θ > dg α, () ṽ(θ β) ṽ(θ α) for vry β n K wth dg β < dg θ, () f γ K and dg γ < dg α, thn ṽ(θ γ) < ṽ(θ α). Dstngushd pars gv rs to dstngushd chans n a natural mannr. A chan θ = θ 0, θ 1,..., θ r of lmnts of K wll b calld a saturatd dstngushd chan for θ f (θ, θ +1 ) s a dstngushd par for 0 r 1 and θ r K. Popscu and Zaharscu (cf. Popscu and Zaharscu, 1995) wr th frst to ntroduc th noton of dstngushd chans. In 1995, thy provd th xstnc of a saturatd dstngushd chan for ach θ blongng to K \ K n cas (K, v) s a complt dscrt rank on valud fld. (cf. In 2005, Aghgh and Khanduja Aghgh and Khanduja, 2005) provd that f (K, v) s a hnslan valud fld of arbtrary rank, thn an lmnt θ blongng to K \ K has a saturatd dstngushd chan wth rspct to v f and only f K(θ) s a dfctlss xtnson of (K, v). A saturatd dstngushd chan for θ gvs rs to svral nvarants assocatd wth θ, som of whch ar gvn by Thorm B statd blow whch s provd n (cf. Aghgh and Khanduja, 2005). Thorm B. Lt (K, v) and ( K, ṽ) b as abov. Lt θ = θ 0, θ 1,..., θ r and θ = η 0, η 1,..., η s b two saturatd dstngushd chans for an lmnt θ blongng to K \ K, thn r = s and [K(θ ) : K] = [K(η ) : K], G(K(θ )) = G(K(η )), K(θ ) = K(η ) for 1 r. Furthr G(K(θ +1 )) G(K(θ )), K(θ +1 ) K(θ ) for 0 r 1. 4

5 In ths papr, w prov Thorm 1.1. Lt v b a hnslan valuaton of arbtrary rank of a fld K wth valu group G. Lt g(x) blongng to R v [x] b a monc polynomal of dgr havng a root θ. Thn for an lmnt a of K, g(x + a) s an Esnstn- Dumas polynomal wth rspct to v f and only f (θ, a) s a dstngushd par and K(θ)/K s a totally ramfd xtnson of dgr. Th followng rsult whch gnralzs a rsult of M. Juras wll b quckly dducd from th abov thorm. Thorm 1.2. Lt g(x) = a x b a monc polynomal wth coffcnts n a hnslan valud fld (K, v). Suppos that th charactrstc of th rsdu fld of v dos not dvd. If thr xsts an lmnt b blongng to K such that g(x + b) s an Esnstn-Dumas polynomal wth rspct to v, thn so s g(x a 1 ). Not that f g(x) blongng to R v [x] s a monc polynomal such that ḡ(x) s rrducbl ovr R v /M v, thn for any non-zro α n M v, g(x) s a Gnralzd Schönmann polynomal wth rspct to f(x) = g(x) α and v. Thrfor to dal wth th scond problm mntond aftr Thorm A, w may assum that ḡ(x) = φ(x) wth φ(x) rrducbl ovr R v /M v and > 1. Whn dg φ(x) = 1 thn th problm rfrrd to abov s alrady solvd n Thorm 1.1 bcaus g(x + a) s an Esnstn-Dumas polynomal wth rspct to v f and only f g(x) s a Gnralzd Schönmann polynomal wth rspct to v and x a. Sttng asd ths two cass, w shall prov Thorm 1.3. Lt v b a hnslan valuaton of arbtrary rank of a fld K wth valu group G and f(x) blongng to R v [x] b a monc polynomal of dgr m > 1 wth f(x) rrducbl ovr th rsdu fld of v. Lt g(x) K[x] b a Gnralzd Schönmann polynomal wth rspct to v and f(x) havng f(x)- 1 xpanson f(x) + g (x)f(x) wth > 1. Lt θ b a root of g(x). Thn for som sutabl root θ 1 of f(x), θ has a saturatd dstngushd chan θ = θ 0, θ 1, θ 2 of lngth 2 wth G(K(θ 1 )) = G, K(θ) = K( θ) and [G(K(θ)) : G] =. Th convrs of th abov rsult s also tru as assrtd by th followng thorm. 5

6 Thorm 1.4. Lt (K, v) b as n th abov thorm. Lt g(x) blongng to R v [x] b a monc polynomal such that ḡ(x) = φ(x), > 1 whr φ(x) s an rrducbl polynomal ovr R v /M v of dgr m > 1. Suppos that a root θ of g(x) has a saturatd dstngushd chan θ = θ 0, θ 1, θ 2 of lngth 2 wth G(K(θ 1 )) = G, K(θ) = K( θ) and [G(K(θ)) : G] =. Thn g(x) s a Gnralzd Schönmann polynomal wth rspct to v and f(x), whr f(x) s th mnmal polynomal of θ 1 ovr K. 2. PROOF OF THEOREMS 1.1, 1.2 Proof of Thorm 1.1. Wrt g(x + a) = x + a 1 x a 0, a, a K. Suppos frst that g(x + a) s an Esnstn-Dumas polynomal wth rspct to v. Thn t s rrducbl ovr K n vw of Thorm A. Snc (K, v) s hnslan, all roots of g(x + a) hav th sam v-valuaton and hnc v(θ a) = v(a 0). In vw of th hypothss that g(x + a) s an Esnstn-Dumas polynomal, w hav = [G + Z v(a 0) : G] = [G + Zv(θ a) : G]. (1) To prov that (θ, a) s a dstngushd par, t s to b shown that max{v(θ β) β K, dg β < dg θ} = v(θ a) = v(a 0). (2) If β s as n (2) and f v(θ β) > v(θ a), thn by th strong trangl law, v(β a) = mn{v(β θ), v(θ a)} = v(θ a) whch n vw of th fundamntal nqualty (cf. Englr and Prstl, 2005, Thorm 3.3.4) and (1) mpls that dg (β a), a contradcton. Thrfor (2) holds and (θ, a) s a dstngushd par wth K(θ)/K totally ramfd n vw of (1). Convrsly suppos that (θ, a) s a dstngushd par and K(θ)/K s a totally ramfd xtnson of dgr. Not that v(θ a) v(θ) 0. Kpng n mnd that (K, v) s hnslan and th rlaton btwn th roots and coffcnts of th th K-rrducbl polynomal g(x + a) = x + a 1 x a 0, w s that v(a ) ( )v(θ a) = ( )v(a 0) 0. So g(x + a) s an Esnstn-Dumas polynomal wth rspct to v onc w show that s v(a 0) / G for any postv numbr s <. Suppos to th contrary thr xsts a postv numbr s < such that 6

7 s v(a 0) = sv(θ a) G, say sv(θ a) = v(b), b K. Snc K(θ)/K s totally ramfd, thr xsts c n K wth v(c) = 0 such that ((θ a) s /b) = c, whch mpls that v((θ a) s bc) > v(b). (3) St v(θ a) = δ and h(x) = (x a) s bc. Lt w dnot th valuaton of K(x) dfnd on K[x] by w( c (x a) ) = mn {ṽ(c ) + δ}, c K. Not that w(h(x)) = mn{sδ, v(bc)} = v(b). Ths qualty wll contradct (3) thrby compltng th proof of th thorm onc w show that To vrfy (4), wrt h(x) = v(h(θ)) = w(h(x)). (4) s (x β () ). Kpng n mnd that h(x) blongng to =1 K[x] s a polynomal of dgr s < and th fact that (θ, a) s a dstngushd par, w hav v(θ β () ) v(θ a) for 1 s and hnc t can b asly sn that v(θ β () ) = mn{v(θ a), v(a β () )} = w(x β () ). On summng ovr, th abov quaton gvs (4). Proof of Thorm 1.2. In vw of Thorm 1.1, t s nough to prov that f (θ, b) s a dstngushd par, thn so s (θ, a 1 ). In fact t suffcs to show that v(θ + a 1 ) v(θ b). (5) Lt θ = θ (1), θ (2),..., θ () dnot th K-conjugats of θ. Usng th hypothss v() = 0, w hav consquntly v(θ + a 1 ) = v(θ + a 1) = v(θ v(θ + a 1 θ () ) = v( =1 (θ θ () )); ) mn {v(θ 2 θ() )} = v(θ θ (2) ) (say). (6) Snc b K, v(θ b) = v(θ (2) b) and hnc v(θ θ (2) ) v(θ b) whch togthr wth (6) provs (5) and hnc th thorm. 7 =2

8 W us Thorm 1.1 to construct xampls of totally ramfd xtnsons K(θ)/K such that no translat of th mnmal polynomal of θ ovr K s an Esnstn-Dumas polynomal wth rspct to v. Notaton. For α sparabl ovr K of dgr > 1, ω K (α) wll stand for th Krasnr s constant dfnd by ω K (α) = max{ṽ(α α ) α α runs ovr K-conjugats of α}. Exampl 2.1. Lt K b th fld of 2-adc numbrs wth th usual valuaton v 2 gvn by v 2 (2) = 1. Th prolongaton of v 2 to th algbrac closur of K wll b dnotd by v 2 agan. Consdr θ = 2+2(2 1/2 )+2 2 (2 1/22 ) and θ 1 = 2+2(2 1/2 ). It wll b shown that K(θ) = K(2 1/4 ) and (θ, θ 1 ) s a dstngushd par. Not that th Krasnr s constant ω K (θ 1 ) = 3/2 and v 2 (θ θ 1 ) = 7/4 > ω K (θ 1 ). Thrfor by Krasnr s Lmma (cf. Englr and Prstl, 2005, Thorm 4.1.7), K(θ 1 ) K(θ) and hnc 2 2 (2 1/4 ) = θ θ 1 blongs to K(θ) as assrtd. To show that (θ, θ 1 ) s a dstngushd par, w frst vrfy that whnvr γ blongng to K satsfs v 2 (θ γ) > v 2 (θ θ 1 ) = 7/4, thn dg γ 4. If γ s as abov, w hav by th strong trangl law v 2 (θ 1 γ) = mn{v 2 (θ 1 θ), v 2 (θ γ)} = 7/4 > ω K (θ 1 ) = 3/2. So by Krasnr s Lmma, K(θ 1 ) K(γ) and hnc G(K(γ)) contans v 2 (θ 1 γ) = 7/4 whch mpls that dg γ 4. Thrfor 7/4 = v 2 (θ θ 1 ) = max{v 2 (θ β) β K, dg β < 4}. Also for any b K wth dg b < dg θ 1, w hav b K and clarly v 2 (θ 1 b) 1/2 < v 2 (θ θ 1 ). So (θ, θ 1 ) s a dstngushd par. As can b asly chckd, θ s a root of g(x) = x 4 8x x 2 80x + 4 whch must b rrducbl ovr K. By Thorm 1.1 no translat of g(x) can b an Esnstn-Dumas polynomal wth rspct to v 2 bcaus (θ, θ 1 ) s a dstngushd par wth θ 1 / K and consquntly (θ, a) cannot b a dstngushd par for any a n K n vw of Thorm B. Morovr, f p b a prm numbr dffrnt from 2, thn no translat of g(x) can b an Esnstn-Dumas polynomal wth rspct to th p-adc valuaton v p, for othrws n vw of Thorm 1.2, g(x + 2) = x 4 4x 2 64x 124 would b an Esnstn-Dumas polynomal wth rspct to v p, whch s clarly mpossbl. 3. PROOF OF THEOREM 1.3. W nd th followng lmma whch s provd n (cf. Brown, 2008, Lmma 4). Its proof s omttd. 8

9 Lmma 3.1. Lt v, G, f(x) and g(x) b as n Thorm A. Lt θ b a root of g(x) and v b a prolongaton of v to K(θ) wth valu group G. Thn v (f(θ)) = v x (g 0 (x)), G = G + Z vx (g 0 (x)), th rsdu fld of v s a smpl xtnson of th rsdu fld of v gnratd by th v -rsdu θ of θ and g(x) s rrducbl ovr K. In partcular th ndx of ramfcaton and th rsdual dgr of v /v ar and dg f(x) rspctvly. Proof of Thorm 1.3. Snc θ s a root of ḡ(x) = f(x) and f(x) s an rrducbl polynomal of dgr m > 1, t follows that v(θ) = 0 and thr xsts a root α () of f(x) such that v(θ α () ) > 0. Lt α b a root of f(x) satsfyng 0 < v(θ α) = max{v(θ α () ) α () runs ovr roots of f(x)} = δ (say). (7) W clam that (θ, α) s a dstngushd par. Obsrv that f γ blongng to K s such that dg γ < dg α, thn v(θ γ) < δ, for othrws by th trangl law w would hav v(α γ) > 0 and hnc ᾱ = γ whch s mpossbl bcaus m = [K(ᾱ) : K] = [K( γ) : K] [K(γ) : K] < m. So to prov th clam, t suffcs to show that whnvr β blongs to K wth v(θ β) > δ, thn dg β dg θ. For provng ths nqualty, n vw of th fundamntal nqualty and th fact dg θ = [G(K(θ)) : G][K(θ) : K] drvd from Lmma 3.1, t s nough to show that G(K(θ)) G(K(β)), K(θ) K(β). (8) Lt β b an lmnt of K wth v(β θ) > δ and α (1), α (2),..., α (m) b th roots of f(x), countd wth multplcts, f any. Wrt m f(β) f(θ) = ( ) β α () = θ α () =1 m =1 ( 1 + β θ ). θ α () Snc v(θ β) > δ and by (7) v(θ α () ) δ for vry, t follows from th abov xprsson for f(β)/f(θ) that ts ṽ-rsdu quals 1 and hnc v(f(β)) = v(f(θ)). Thrfor n vw of Lmma 3.1, G(K(θ)) = G + Zv(f(θ)) G(K(β)). Also kpng n mnd that v(θ β) > δ > 0, w hav by Lmma 3.1, K(θ) = K( θ) = K( β) whch provs (8) and hnc th clam. Rcall that ᾱ s a root of th rrducbl polynomal f(x) of dgr m > 1. So v(α 1) = 0. Also for any β n K wth dg β < dg α, w hav v(α β) 0, 9

10 for othrws ᾱ = β and ths n vw of th fundamntal nqualty would mply [K(β) : K] [K( β) : K] = m. So (α, 1) s a dstngushd par. Thus w hav provd that θ has a saturatd dstngushd chan θ, α, 1 of lngth 2. Snc [K(α) : K] = [K(ᾱ) : K] = m, t follows from th fundamntal nqualty that G(K(α)) = G. Th othr two qualts hold by vrtu of Lmma PROOF OF THEOREM 1.4. W rtan th notatons of th prvous sctons as wll as th assumpton that v s a hnslan valuaton of arbtrary rank of a fld K wth unqu prolongaton ṽ to th algbrac closur K havng valu group G. Rcall that a par (α, δ) blongng to K G s sad to b a mnmal par (mor prcsly (K, v)-mnmal par) f whnvr β blongng to K satsfs ṽ(α β) δ, thn dg β dg α. It can b asly sn that f (θ, α) s a dstngushd par and δ = ṽ(θ α), thn (α, δ) s a mnmal par. Lt (α, δ) b a (K, v)-mnmal par. Th valuaton w α,δ of K(x) dfnd on K[x] by w α,δ ( c (x α) ) = mn{ṽ(c ) + δ}, c K wll b rfrrd to as th valuaton dfnd by th par (α, δ). Th dscrpton of w α,δ on K[x] s gvn by th alrady known thorm statd blow (cf. Alxandru, Popscu and Zaharscu, 1988; Khanduja, 1992). Thorm C. Lt w α,δ b th valuaton of K(x) dfnd by a mnmal par (α, δ) and w α,δ b th valuaton of K(x) obtand by rstrctng w α,δ. Lt f(x) b th mnmal polynomal of α ovr K. Thn for any polynomal g(x) n K[x] wth f(x)-xpanson g (x)f(x), on has w α,δ (g(x)) = mn{ṽ(g (α))+w α,δ (f(x))}. 0 Th followng rsult provd n (cf. Aghgh and Khanduja, 2005, Thorm 1.1()) wll b usd n th squl. Lmma D. Lt (θ, α) b a (K, v)-dstngushd par and f(x) b th mnmal polynomal of α ovr K. Thn G(K(θ)) = G(K(α)) + Zv(f(θ)). Proof of Thorm 1.4. W dvd th proof nto thr stps. Stp I. Lt f(x) b th mnmal polynomal of θ 1 ovr K. In ths stp, w prov that f(x) s rrducbl ovr K and f(x) = φ(x). In vw of th hypothss [K(θ) : K] = m, [G(K(θ)) : G] = and th fundamntal nqualty, t follows that [K(θ) : K] m. Snc θ s a root of th polynomal g(x) havng dgr m, 10

11 w hav [K(θ) : K] = m. Not that v(θ θ 1 ) > 0, bcaus f F (x) R v [x] s a monc polynomal wth F (x) = φ(x), thn thr xsts a root β of F (x) such that β = θ whch n vw of th hypothss > 1 mpls that v(θ θ 1 ) v(θ β) > 0. So th assrton of Stp I s provd onc w show that [K(θ 1 ) : K] = [K( θ 1 ) : K] = m. (9) Rcall that K(θ 1 ) K(θ) by Thorm B. Thrfor usng th hypothss K(θ) = K( θ) and th fact θ 1 = θ, t follows that K(θ 1 ) = K( θ); n partcular [K(θ 1 ) : K] = [K( θ 1 ) : K] = m. (10) Snc K(θ 1 )/K s a dfctlss xtnson n vw of (Aghgh and Khanduja, 2005, Thorm 1.2) and t s gvn that G(K(θ 1 )) = G, w now obtan (9) usng (10). Stp II. For smplcty of notaton, w shall hncforth dnot θ 1 by α. St 1 v(θ α) = δ and v(f(θ)) = λ. Lt g(x) = f(x) + g (x)f(x) b th f(x)- xpanson of g(x). Lt w α,δ b th valuaton of K(x) dfnd by th mnmal par (α, δ). In ths stp, w prov that and Wrt f(x) = m (x α () ), g(x) = =1 w α,δ (f(x)) = λ (11) w α,δ (g(x)) = v x (g 0 (x)) = λ. (12) m j=1 and hnc v(θ α () ) = mn{δ, v(α α () )}, w hav (x θ (j) ). Usng th fact that v(θ α () ) δ m w α,δ (f(x)) = w α,δ ( (x α () )) = =1 m mn{δ, v(α α () )} = =1 m v(θ α () ) = λ =1 whch provs (11). Snc (K, v) s hnslan, for any K-conjugat θ (j) of θ, thr xsts a K-conjugat α () of α such that v(θ (j) α) = v(θ α () ) δ; consquntly w α,δ (x θ (j) ) = mn{δ, v(α θ (j) )} = v(α θ (j) ), whch on summng ovr j gvs w α,δ (g(x)) = v(g(α)). (13) 11

12 Rcall that n vw of Stp I, f(x) s rrducbl ovr K of dgr m havng θ as a root. So for any polynomal A(x)= a x blongng to K[x] of dgr lss than m, w hav v(a(θ)) = v x (A(x)) = mn {v(a )}, (14) for f th abov qualty dos not hold, thn m > 1, v(θ) = 0 and hnc th trangl law would mply v(a(θ)) > mn{v(a θ )} = v(a j ) (say) and thus m 1 (a /a j ) θ = 0, whch s mpossbl. Kpng n mnd th f(x)-xpanson of g(x) and that f(α) = 0, w s that g(α) = g 0 (α) and consquntly t follows from (14) that v(g(α)) = v(g 0 (α)) = v x (g 0 (x)) whch togthr wth (13) provs th frst qualty of (12). As f(x), g(x) ar rrducbl ovr th hnslan valud fld (K, v), w hav v(f(θ (j) )) = v(f(θ)), 1 j m, v(g(α () )) = v(g(α)), 1 m. (15) Kpng n mnd that m g(α () ) = ± =1 m j=1 f(θ (j) ), t s clar from (15) that v(g(α)) = v(f(θ)) = λ, whch provs th scond qualty of (12) n vw of (13). Stp III. In ths stp, w prov that g(x) s a Gnralzd Schönmann polynomal wth rspct to v and f(x). By Thorm C, (11) and (12), w hav w α,δ (g(x)) = mn 0 {v(g (α)) + λ} = v x (g 0 (x)) = λ. (16) As v(g (α)) = v x (g (x)), (16) shows that v x (g (x)) + λ λ for 0 1,.., v x (g (x)) λ = vx (g 0 (x)) > 0. Rcall that λ = v(f(θ)). Snc (θ, α) s a dstngushd par, n vw of Lmma D and th hypothss G(K(α)) = G, w hav G(K(θ)) = G + Zλ. (17) By hypothss [G(K(θ)) : G] =, so t follows from (17) that s th smallst postv ntgr for whch λ G. Ths complts th proof of th thorm. Exampl 4.1. Lt K b th fld of 3-adc numbrs wth th usual valuaton v 3 whos xtnson to th algbrac closur K of K wll b dnotd by ṽ 3. 12

13 Consdr th polynomal g(x) = x x wth ḡ(x) = (x 2 + 1) 2. It can b asly sn that θ = (2 + 3) s a root of g(x) whr = 1. Snc θ = 2 / K and ṽ 3 (θ 2 2) = 1/2, t follows n vw of th fundamntal nqualty that [K(θ) : K] = 4. A smpl calculaton shows that th Krasnr s constant ω K (θ) = ṽ 3 (θ 2) = 1/2. So by Krasnr s Lmma, ṽ 3 (θ β) 1 2 for vry β K wth dg β < 4. Furthr f for som γ n K, ṽ 3 (θ γ) = 1 2, thn θ = γ = 2. Snc 2 / K, w s that [K(γ) : K] 2. Thrfor (θ, 2) s a dstngushd par. It can b asly sn that (2, 0) s a dstngushd par and hnc θ, 2, 0 s a saturatd dstngushd chan for θ satsfyng th hypothss of Thorm 1.4. So g(x) s a Gnralzd Schönmann polynomal wth rspct to v 3 and f(x) = x ACKNOWLEDGEMENTS Th fnancal support by CSIR (grant no. 09/135(0539)/2008-EMR-I) and Natonal Board for Hghr Mathmatcs, Mumba s gratfully acknowldgd. REFERENCES Aghgh, K., Khanduja, S. K. (2005). On chans assocatd wth lmnts algbrac ovr a hnslan valud fld. Algbra Colloq. 12(4): Alxandru, V., Popscu, N., Zaharscu, A. (1988). A thorm of charactrzaton of rsdual transcndntal xtnsons of a valuaton. J. Math. Kyoto Unv 28(4): Brown, R. (2008). Roots of gnralzd Schönmann polynomals n hnslan xtnson flds. Indan J. Pur Appl. Math. 39(5): Englr, A. J., Prstl, A. (2005). Valud Flds. Brln: Sprngr-Vrlag. Juráš, M. (2006). Esnstn-Dumas crtron for rrducblty of polynomals and projctv transformatons of th ndpndnt varabl. JP J. Algbra, Numbr Thory and Applcatons 6(2): Khanduja, S. K. (1992). On valuatons of K(x). Proc. Ednb. Math. Soc. 35: Khanduja, S. K., Saha, J. (1997). On a gnralzaton of Esnstn s rrducblty crtron. Mathmatka 44: Popscu, N., Zaharscu, A. (1995). On th structur of th rrducbl polynomals ovr local flds. J. Numbr Thory 52: Rbnbom, P. (1999). Th Thory of Classcal Valuatons. Nw York: Sprngr- Vrlag. 13