ON RAMIFICATION POLYGONS AND GALOIS GROUPS OF EISENSTEIN POLYNOMIALS

Transcription

1 ON RAMIFICATION POLYGONS AND GALOIS GROUPS OF EISENSTEIN POLYNOMIALS CHRISTIAN GREVE AND SEBASTIAN PAULI Abstract Lt ϕx b an Esnstn polynomal of dgr n ovr a local fld and α b a root of ϕx Our man tool s th ramfcaton polygon of ϕx, that s th Nwton polygon of ρx = ϕαx + α/α n x W prsnt a mthod for dtrmnng th Galos group of ϕx n th cas whr th ramfcaton polygon conssts of on sgmnt 1 Introducton Algorthms for computng Galos groups ar an mportant tool n constructv numbr thory Th commonly usd algorthms for computng Galos groups of polynomals ovr Q and Qt ar basd on th mthod of Stauduhar [20] Consdrabl progrss has bn mad n ths ara ovr th last tn yars For local flds howvr, that s for flds K complt wth rspct to a non-archmdan xponntal valuaton ν wth rsdu class fld K of charactrstc p, thr s no gnral algorthm As th Galos groups of unramfd xtnsons ar xplctly known, w concntrat th Galos groups of totally ramfd xtnsons Ths can b gnratd by an Esnstn polynomal, that s a polynomal ϕx = x n + n 1 =0 ϕ x O K [x] wth νϕ 0 = 1 and νϕ 1 for 1 n 1 W dnot by α a root of ϕx n an algbrac closur K of K If p dos not dvd th dgr of ϕx th xtnson Kα s tamly ramfd and can b gnratd by a pur polynomal W show how ths pur polynomal can b obtand and rcall th wll known xplct dscrpton of ts Galos group s scton 2 If p dvds th dgr of ϕx th stuaton bcoms mor dffcult John Jons and Davd Robrts hav dvlopd algorthms for dtrmnng th Galos group for th spcal cass of polynomals of dgr 2 2, 2 3, and 3 2 ovr Q 2 and Q 3 rspctvly basd on th rsolvnt mthod [7, 8, 9] A usful tool for obtanng nformaton about th splttng fld and th Galos group s th ramfcaton polygon rpϕ of ϕx, whch s th Nwton polygon npρ of th ramfcaton polynomal ρx = ϕαx + α/α n x Kα[x] Davd Romano has tratd th cas of Esnstn polynomals ϕx of dgr n = p m wth νϕ 1 = 1, so that th ramfcaton polygon conssts of on sgmnt of slop h/p m 1 whr gcdh, p m 1 = 1 [16] In ths cas th Galos group of ϕx s somorphc to th group Γ = { x ax σ + b a F p m, b F p m, σ GalF p m/f p m K } of prmutatons of F p m In [17] h gnralzs hs rsult to ramfcaton polygons that ar a ln, on whch only th nd ponts hav ntgral coordnats W dvlop th thory of ramfcaton polygons furthr by attachng an addtonal nvarant, th assocatd nrta to ach sgmnt of th ramfcaton polygon scton 3 In scton 4 w dscrb th shap of ramfcaton polygons and show that th polygons, as wll as th 1

2 assocatd nrtas, whch ar th dgrs of th splttng flds of assocatd polynomals, ar nvarants of Kα W fnd that th sgmnts of th ramfcaton polygon corrsponds to th subflds of th fld gnratd by ϕx scton 5 In scton 6 w nvstgat how ramfcaton polygons and thr assocatd nrtas bhav n towrs of subflds Ths s followd by a dscrpton of th splttng fld n th cas of ramfcaton polygons wth on sgmnt and a dscrpton of th maxmal tamly ramfd subfld of th splttng fld n th gnral cas scton 7 In scton 8 w us ths rsults to fnd Galos groups of Esnstn polynomals wth on sdd ramfcaton polygon wth arbtrary slops Our mthods can b gnralzd algorthmcally to th cas of ramfcaton polygons wth two sgmnts [2] and byond In scton 9 w gv som xampls for Galos groups dtrmnd n ths way Notaton In th followng K s a local fld, complt wth rspct to a non-archmdan xponntal valuaton ν, whr ν = ν K s normalzd such that νπ = ν K π = 1 for a unformzng lmnt π n th valuaton rng O K of K Th contnuaton of ν to an algbrac closur K of K s also dnotd by ν For γ K and γ K w wrt γ γ, f νγ γ > νγ W wrt ζ n for a prmtv n-th root of unty W dnot by K = O K /π = F q th rsdu class fld of O K and by β = β + π th class of β O K n K For γ K w dnot by γ a lft of γ to O K If ϕx = n =1 ϕ x O K [x] w st ϕx := n =1 ϕ x K[x] 2 Tamly Ramfd Extnsons W prsnt som rsults about tamly ramfd subflds of totally ramfd xtnsons and th splttng flds and Galos groups of totally and tamly ramfd xtnsons Proposton 21 Lt n = 0 p m wth p 0 and lt n 1 ϕx = x n + ϕ x + ϕ 0 O K [x] =1 b a polynomal whos Nwton polygon s a ln of slop h/n, whr gcdh, n = 1 Lt α b a root of ϕx Th maxmum tamly ramfd subxtnson M of L = Kα of dgr 0 can b gnratd by th Esnstn polynomal x 0 + ψ b 0π 0a wth ψ 0 ϕ 0 mod π h+1 and whr a and b ar ntgrs such that a 0 + bh = 1 Proof As th Nwton polygon of ϕx s a ln all roots α of ϕx hav th sam valuaton, namly να = h/n Bcaus gcdh, n = 1, for ach root α of ϕx, th xtnson Kα/K s totally ramfd of dgr n, whch mpls that ϕx s rrducbl Snc n = 0 p m wth gcd 0, p = 1 ts maxmum tamly ramfd subxtnson M ovr K has dgr [M : K] = 0 W frst show that M and th xtnsons gnratd by x 0 + ψ 0 ar somorphc Bcaus νϕ 0 = h and ψ 0 ϕ 0 mod π h+1, thr s a prncpal unt 1+πε O K such that ψ 0 = 1+πεϕ 0 Furthrmor α n = ϕ 0 n 1 =1 ϕ α = 1+π L δϕ 0 for som prncpal unt 1 + π L δ O L whr π L s a unformzr of th valuaton rng O L of L Th polynomal x 0 + ψ 0 has a root ovr L f and only f α pm x 0 + ψ 0 has a root ovr L Dvson by α n ylds x 0 + ψ 0 α n = x πεϕ π L δϕ 0 x 0 1 mod π L O L [x] Obvously ρx = x 1 L[x] s squar fr and ρ1 = 0 Wth Nwton lftng and by rvrsng th transformatons abov w obtan a root of x 0 + ψ 0 n L Lt β b ths root of 2

3 x 0 + ψ 0 Thn νβ b π a = bh/ + a = 1/ and M = Kβ = Kβ b π a As β 0b π 0a = ψ0π b 0a w hav β b π a s a root of x 0 + ψ0π b 0a O K [x] Corollary 22 Lt ϕx = =0 ϕ x O K [x] b an Esnstn polynomal and assum p If ψx = x + ψ 0 wth ψ 0 ϕ 0 mod π 2, thn th xtnsons gnratd by ϕx and ψx ar somorphc It follows from Corollary 22 that th splttng fld of an Esnstn polynomal ϕx = =0 ϕ x O K [x] wth p s N = Kζ, ϕ 0, whr ζ s a prmtv -th root of unty Th Galos group of N/K s wll known, w obtan t from th gnral dscrpton of Galos groups of normal, tamly ramfd xtnsons s, for nstanc, [5, chaptr 16]: Thorm 23 Lt K b a local fld and q th numbr of lmnts of ts rsdu class fld Lt N/K b a normal, tamly ramfd xtnson wth ramfcaton ndx and nrta dgr f Thr xsts an ntgr r wth rq 1 0 mod such that N = Kζ, ζ r π, whr ζ s a q f 1-st root of unty and q f 1 0 mod Lt k = rq 1 Th gnrators of th Galos group ar th automorphsms s : ζ ζ, ζ r π ζ qf 1/ ζ r π and t : ζ ζ q, ζ r π ζ k ζ r π Th Galos group of N/K as a fntly prsntd group s s, t s = 1, t f = s r, s t = s q = C C f, 3 Assocatd Polynomals Assocatd or rsdual polynomals, frst ntroducd by Or [14], ar a usful tool n th computaton of dal dcompostons and ntgral bass [11, 12, 3] and th closly rlatd problm of polynomal factorzaton ovr local flds [4, 15] Th assocatd polynomals yld nformaton about th unramfd part of th xtnson gnratd by a polynomal W wll us t n th constructon of splttng flds of Esnstn polynomals Lt ρx = n =0 ρ x O K [x] b a not ncssarly rrducbl monc polynomal whos Nwton polygon npρ conssts of t sgmnts: 0, νρ 0 a 1, νρ a1,, a t 1, νρ at 1 a t, νρ at wth slops: h 1 / 1 < h 2 / 2 < < h t 1 / t 1 < h t / t wth gcd, h = 1 for 1 t Each of th sgmnts corrsponds to a factor ρ r x of ρx For ach sgmnt w obtan on assocatd polynomal as follows For 1 r t lt b r = νρ ar Consdr th r-th sgmnt a r 1, b r 1 a r, b r of npρ and st d r = a r a r 1 W hav νρar νρa r 1 d r = hr r Lt β b a root of ρx wth νβ = h r / r, st L = Kβ, lt π L b an unformzng lmnt n th valuaton rng O L of L W hav ρβx π b r 1 β a r 1 = n ρ β x π b r 1 β a r 1 =0 a r =a r 1 d r/ r j=0 ρ βx π b r 1 β a r 1 mod π L O L [x] ρ j+ar 1 β jr+a r 1 x jr+a r 1 3 π b r 1 β a r 1 mod π L O L [x]

4 Th last congrunc holds, bcaus th x-coordnats of th ponts on th r-th sgmnt of th Nwton polygon ar of th form a r 1 + j r 0 j a r 1 a r / r Dvson by x a r 1 ylds ρβx π b r 1 β a r 1x a r 1 d r/ r j=0 ρ j+ar 1 β jr x jr π b r 1 mod π L O L [x] For γ = β r /π hr w hav νγ = νβ r /π hr = 0 By substtutng γπ hr for β r w gt ρβx π b r 1 β a r 1x a r 1 d r/ r j=0 ρ jr+a r 1 π jhr γx r j π b r 1 mod π L O L [x] If w rplac γx r by y w obtan th assocatd polynomal of ρx wth rspct to th r-th sgmnt S r of npρ: A r y := d r/ r j=0 It follows mmdatly from th constructon that: ρ jr+a r 1 π jhr b r 1 y j K[y] Lmma 31 Lt β 1,, β n b th roots of ρx Th roots of Ay K[y] ar of th form β j π h j for som 1 n and som 1 j t Dfnton 32 Lt Ay K[y] b th assocatd polynomal of a sgmnt S of npρ and γ a root of Ay W call th dgr of th splttng fld of A r y K[y] ovr K th assocatd nrta of S Rmark 33 Th dnomnators of th slops n lowst trms of th sgmnts of th Nwton polygon npρ of a polynomal ρx ar dvsors of th ramfcaton ndcs of th xtnsons gnratd by th rrducbl factors of a polynomal For ach sgmnt of npρ th assocatd nrta s a dvsor of th nrta dgr of ths xtnsons Rmark 34 Th factorzaton of A r y ylds a factorzaton of th factor of ρx that corrsponds to S r [14] 4 Ramfcaton Polygons Lt ϕx = n =0 ϕ x O K [x] b an Esnstn polynomal, α K a root of ϕx and L := Kα Th polynomal ρx = n ρ x := =0 ϕαx + α α n x O L [x] s calld th ramfcaton polynomal of ϕx and ts Nwton polygon, whch w dnot by rpϕ, s calld th ramfcaton polygon of ϕx also s [18] Dnotng th roots of ϕx n K by α = α 1,, α n w hav ρx = n =2 x α α = α 4 n =2 x + 1 α α

5 If th xtnson L/K gnratd by ϕx s Galos wth Galos group G th sgmnts of th ramfcaton polygon rpϕ corrspond to th ramfcaton subgroups of G: G j := {σ G ν L σα α j + 1} for j 0 Bcaus ν L α α α = ν Lα α 1 th ramfcaton polygon dscrbs th fltraton G = G 0 G 1 G k = 1 of th Galos group, that s, a sgmnt of slop m ylds a jump at m n th fltraton, whch mans G m G m+1 If th xtnson L/K s not Galos, thr s a smlar ntrprtaton for a fltraton of th st of mbddngs of L/K n K n th contxt of non-galos ramfcaton thory s [6] From th nxt lmma on can dduc th typcal shap of th ramfcaton polygon s fgur 1 Lmma 41 [18, Lmma 1] Lt ϕx = n =0 ϕ x K[x] b an Esnstn polynomal and n = 0 p m wth p 0 Dnot by α a root of ϕx and st L = Kα Thn th followng hold for th coffcnts of th polynomal ψx = n =0 ψ x := ϕαx + α L[x]: a ν L ψ n for all b ν L ψ p m = ν L ψ n = n c ν L ψ ν L ψ p s for p s < p s+1 and s < m ν L ρ m 1 m 2 m l 0 p s 1 1 p s 2 1 p s l 1 1 p s l 1 n 1 As a consqunc w hav: Fgur 1 Shap of th ramfcaton polygon Lmma 42 If a ramfcaton polygon conssts of only on sgmnt wth slop, say m 1 thn m 1 pνp p 1 Proposton & Dfnton 43 Lt L/K b totally ramfd and α a prm lmnt of L and ϕx th mnmal polynomal of α Thn rpϕ and th assocatd nrta of th sgmnts ar nvarants of L/K W call rpl/k := rpϕ th ramfcaton polygon of L/K Proof Lt β = δα whr δ O L = O K [α] wth ν L δ = 0 W can wrt δ n th form δ = δ 0 + δ 1 α + δ 2 α 2 + wth δ O K Lt β = β 1,, β n b th conjugats of β and lt 5

6 ϕx b th mnmal polynomal of β W compar th roots of th ramfcaton polynomals of ϕx and ϕx n ρx = x α α n = x 1 + α α α and ρx = =2 n =2 For 1 n long dvson ylds x β β = β =2 n =2 x 1 + β β β β = δ 0α + δ 1 α 2 + δ 0 α + δ 1 α 2 + = α α + δ 1α αα + δ 0 α + δ 1 α 2 + W hav ν L 1+α /α = m whr m s on of th slops of rpϕ As ν L α αα = m+2 w hav 1 β /β 1 α /α Thus ν L β β = m + 1 and t follows that th slops of th ramfcaton polygon ar ndpndnt of th choc of th unformzng lmnt of L and thrfor nvarants of L To prov that th assocatd nrta s an nvarant of L/K w consdr th sgmnt wth slop m = h/ of th Nwton polygons of ρx and ρx Th roots of th corrspondng assocatd polynomals Ay L[y] and Ãy L[y] wth rspct to th sgmnt wth slop m ar of th form s Lmma 31: 1 + α /α 1 + β /β and α h Bcaus 1 + β /β 1 + α /α w hav 1 + β /β β h β h 1 δ h 1 + α /α α h Thrfor th roots of Ay and Ãy dffr only by th factor δ h K = K So, f Ay = d =1 y γ thn Ãy = d =1 y γ undrlnδ h Clarly th polynomals Ay and Ãy hav th sam splttng flds whch mpls that th assocatd nrtas ar th sam Lmma 44 Lt L/K b totally ramfd of dgr p m and lt m 1,, m l b th slops of rpl/k Lt T/K b tamly ramfd wth ramfcaton ndx 0 and N = TK Thn th slops of rpn/t ar 0 m 1,, 0 m l Proof Lt α b a unformzr of L/K, ϕx ts mnmal polynomal and α = α 1,, α p m K ts conjugats Lt β b a unformzng lmnt of T If a, b Z such that a 0 bp m = 1 thn ν T α a /β b = 1/p m Th ramfcaton polynomal ρx O N [x] of th mnmal polynomal of α a /β b s ρx = p m =2 x + 1 αa β b = β b α a p m =2 x + 1 αa α a Each quotnt α /α s of th form 1 + γ α mq wth νγ = 0 for som 1 q l As gcda, p = 1 w gt α /α a = 1 + γα mq a 1 + aγα mq, whch mpls that th xponntal 6

7 valuaton of th roots of ρx T[x] ar m q 1 q l Thus th slops of rpn/t ar 0 m 1,, 0 m l 5 Blocks and Subflds In th followng w us th conncton btwn blocks of th Galos group and subflds of an xtnson to dscrb and calculat a spcfc chan of subflds of our xtnson L/K, whch s not Galos n gnral W dnot by G = Galϕ = GalL/K th Galos group of L/K, whch s th automorphsm group of th normal closur of L ovr K As ϕx s rrducbl, G acts transtvly on th st of roots Ω = {α 1,, α n } of ϕx Dfnton 51 A non-mpty subst of Ω s calld a block, f σ {, } for all σ G Th group G := {σ G σ = } s calld th stablzr of Th st { = 1,, k } := {σ σ G} s th block systm wth rspct to It consttuts a partton of Ω, thus n = k For th rmandr of ths scton w fx th followng notaton compar fgur 1 Th Esnstn polynomal ϕx has dgr n = 0 p m, ts ramfcaton polynomal s dnotd by ρx = n 1 j=0 ρ jx j, and ts ramfcaton polygon rpϕ conssts of l + 1 sgmnts By Lmma 41 thr ar natural numbrs 0 = s 0 < s 1 < < s l = r, so that th -th sgmnt S s of th form p s 1 1, ν L ρ p s 1 1 p s 1, ν L ρ p s 1 for 1 l Th last sgmnt S l+1 = p l 1, 0 n 1, 0 s horzontal W dnot th slops of th sgmnt of rpϕ by m 1 < m 2 < < m l+1 = 0 W choos th numbrng of th roots α = α 1,, α n of ϕx compatbl to th ramfcaton polygon, that s, such that, for 1 l + 1 ν L αp s 1 +1 α 1 α 1 = = ν L αp s α 1 α 1 = m Th followng lmma says, that w can rfn ths numbrng accordng to crtan block systms Lmma 52 Th Galos group of ϕx has th blocks = {α 1,, α p s } = {α K ϕα = 0 and ν L α α 1 m + 1} W can ordr th roots α 1,, α n such that r and k = n/p s 1 l = { α r 1p s +1,, α rp s } for 1 r k Proof Lt σ Galϕ W show, that σ s mpty or qual to If σα 1 w hav ν L σα 1 α 1 m + 1 Thn w hav for an arbtrary α j that ν L σα j α 1 = ν L σα j σα 1 + σα 1 α 1 = ν L σα j α 1 + σα 1 α 1 m + 1, snc ν L σα j α 1 m + 1 Bcaus of our choc of ordrng of α 1,, α n only th dffrncs α k α 1 for k p s hav xponntal valuaton gratr than or qual to m + 1, whch mpls σα j 7

8 L 0 = L = Kα 1 0 = {α 1 } p s 1 L 1 = Kα 1 α p s 1 1 = {α 1,, α p s 1 } p s 2 s 1 p s l 1 s l 2 L l 1 = Kα 1 α p s l 1 l 1 = {α 1,, α p s l 1 } p s l s l 1 L l = Kα 1 α p s l l = {α 1,, α p s l} 0 L l+1 = K l+1 = {α 1,, α n } Fgur 2 Th subflds of L = Kα 1 and th corrspondng blocks, whr th roots of α 1,, α n of ϕx O K [x] ar ordrd as n Lmma 52 and n = 0 p l wth p 0 If σα 1 / w hav ν L σα 1 α 1 < m + 1 In ths cas w gt for an arbtrary α j, that ν L σα j α 1 = ν L σα j α 1 + σα 1 α 1 < m + 1 and σα j / follows Th ordrng of th roots α accordng to th ramfcaton polygon and th subordrng accordng to th block systms { 1,, k } ar consstnt, bcaus 1 2 l Thr s a corrspondnc btwn th blocks and th subflds of L/K For a subgroup H of th Galos group G of L/K w wrt FxH for th fxd fld undr H A proof of th thorm can b found n [10] Thorm 53 Lt ϕx K[x] b rrducbl of dgr n, ϕα = 0, L = Kα, and G th Galos group of L/K a Th corrspondnc FxG s a bjcton btwn th st of blocks contanng α and th st of subflds of L/K b For two blocks 1, 2 wth corrspondng subflds L 1, L 2 w hav L 1 L 2 f and only f 2 1 Th nxt thorm dscrbs th subflds corrspondng to th blocks of Lmma 52, s Fgur 2 Thorm 54 Lt L = Kα and for 0 l lt L = Kβ wth β = α 1 α p s Thn L = L 0 L 1 L l K wth [L : L +1 ] = p s +1 s for l 1 and [L l : K] = Proof For 1 l dnot by E th fld FxG that corrsponds to th block = {α 1,, α p s } s Thorm 53 Th fld E has dgr m = n ovr K W show, that 8 p s

9 E = L Lt hx = α j x α j Bcaus hx stays nvarant undr th acton of G and t dvds ϕx and t has th corrct dgr p s, t s th mnmal polynomal of α 1 ovr E Th fld gnratd by th coffcnts of hx s qual to E If ths fld had dgr m < m ovr K, w would gt th contradcton [L : K] = m p s < mp s = n In fact, vn th constant coffcnt β of hx s suffcnt to gnrat E, snc L/K s totally ramfd and ν L β = p s = [L : E ] holds Hnc β s a prmtv lmnt for E /K and w gt E = L as proposd Th ncluson and dgr statmnts follow drctly from thorm 53 In th cas of a Galos xtnson L/K th subflds L ar xactly th ramfcaton subflds of L/K, that s, th fxd flds undr th ramfcaton subgroups of th Galos group For an xtnson, whch s not Galos, thy ar fxd flds undr th ramfcaton substs of th st of mbddngs of L/K n K s agan [6] Rmark 55 An xplct computaton of th subflds L = Kβ on a computr mans fndng th lmnts β as lmnts of L/K Ths ylds an mbddng of L /K n L/K Thn a gnratng polynomal for L /K can b obtand by a mnmal polynomal calculaton W brfly dscrb how th lmnts β n L/K can b dtrmnd Polynomal factorzaton tchnqus [1, 15] yld a factorzaton ρx = r 1 x r l+1 x L[x] of th ramfcaton polynomal, whr r j x corrsponds to th j-th sgmnt of th ramfcaton polygon Rvrsng th transformaton from ϕx to ρx ths gvs us a factorzaton ϕx = p 1 x p l+1 x L[x] of ϕx wth p j x = x α K[x] for 2 j l + 1 and p 1 x = x α ν L α α α =m j ν L α α α =m 1 x α K[x] Now γ L s qual to th constant coffcnt of p 1 x p x 6 Ramfcaton Polygons and Subflds W nvstgat how ramfcaton polygons and thr assocatd nrtas bhav n towrs of subflds W contnu to us th notaton from scton 5 also s Fgur 2 By constructon of th subflds L w xpct a strong conncton btwn th ramfcaton polygon of ϕx and th ramfcaton polygons of gnratng polynomals for th xtnsons L 1 /L Th followng lmma and thorm dscrb th conncton n dtal Lmma 61 Assum th ramfcaton polygon of rpϕ = rpl/k conssts of th sgmnts S 1,, S l+1 of lngths p s 1 1, p s 2 p s 1,, p s l+1 p s l wth slops m1 < < m l+1 = 0 Thn a th ramfcaton polygon rpl 1 /K has xactly l sgmnts T 1,, T l of lngths p s 2 /p s 1 1, p s 3 p s 2 /p s 1, p s l+1 p s l/p s 1 wth slops m 2,, m l+1 = 0, b th assocatd nrta of T s qual to th assocatd nrta of S +1, and c for ach root δ of A y th lmnt δ ps 1 s a root of th assocatd polynomal of T 9

10 Proof W assum that th roots of ϕx ar ordrd as n Lmma 52 Lt 1 = 1 1,, k 1 b th block systm for th smallst block 1 If α r 1 wth 2 r k, thn ν L α α 1 = m λ +1 < m 1 +1 for som λ {2,, l+1} Thus w can wrt α = α 1 +δα m λ+1 1 for som δ K wth νδ = 0 If α, α r 1 thn ν L α α = m Thus, f α = α 1 + δ α m λ+1 1 for som δ K wth νδ = 0 thn δ δ Rcall that by our ordrng of th roots of ϕx w hav α 1 1 and α r 1p s 1 + r 1 for 1 p s 1 and 2 r k By th consdratons abov thr s a ε K wth νε = 0 so that for som λ {2,, l + 1}: α r 1p s 1 + α εα m λ 1 For 1 r k lt β r = α, so that L α r 1 = Kβ 1 Thn ψx = k r 1 x β r s th 1 mnmal polynomal of β 1 ovr K Th ramfcaton polynomal of ψx s: ψβ 1 x + β 1 β k 1 x = k r=2 x 1 + β r β 1 = k r=2 x 1 + α r 1p s 1 +1 α rp s 1 α 1 α p s 1 By rlaton 1 thr ar ε r K wth νε r = 0 and λ {2,, l + 1} so that 1 + β p s 1 1 r ε r α m λ 1 ps 1 = ε ps 1 r α m λp s 1 p s ε β 1 rα m λ 1 If w show that β r β 1 ε ps1 α m λps1 1, =1 thn clarly a holds W now proof b and c for S 2 and T 1 Th rsults for th othr sgmnts follow analogously Th roots of th ramfcaton polygon of ϕ wth valuaton m 2 ar 1 + α /α 1 ε α m 2 1 for som ε K wth νε = 0 and p s p s 2 By Lmma 31 ths gvs th roots ε α m = ε 2 α h 2 1 of th assocatd polynomal A 2 yk[y] of S 2, whr m 2 = h2/ 2 wth gcdh 2, 2 = 1 For ach ε 2 of A 2 y thr s root of th ramfcaton polynomal of ψx wth 1 + β r /β 1 s 1 α mp 2 ε ps 1 1 Wth ths w obtan th corrspondng roots of th assocatd polynomal B 1 y K[y] of T 1 : ε ps 1 α m 2p s β h 2 1 = ε 2p s 1 α h 2p s α 1 α p s 1 h 2 = ε 2 p s 1 Ths proofs c As ε ε p s an automorphsm of K th splttng flds of A 2 y and B 1 y ar somorphc, whch mpls b To proof rlaton 2 w nd to show that ps1 1 ν L =1 p s 1 ε α m λ 10 > m λ p s 1

11 By th ultramtrc nqualty t s suffcnt to show that for 1 p s 1 1 p s 1 ν L γ α m λ > m λ p s 1 As ν p s 1 p = s1 ν p ths smplfs to ν L ps 1 ν p + m λ > m λ p s 1 or By Lmma 42 w hav whch s quvalnt to ν L ps 1 ν p p s 1 > m λ ν L p p s 1 1 p 1 m 1 > m λ So t s suffcnt to show that ν L ps 1 ν p p s 1 ν L p p s 1 1 p 1, pp s 1 p s 1 p 1s1 ν p 1 W wrt = ap v wth p a and v < s 1 and obtan pp s 1 p s 1 p 1s1 ν p Ths complts th proof = = pp s 1 p v p s 1 p 1s1 v = p p 1 p s 1 v 1 p s 1 v s 1 v p p 1 1 1/ps1 v = 1 1/ps 1 v 1 s 1 v 1 1/p s 1 v p p s 1 v 1 s 1 v 1 Thorm 62 Th ramfcaton polygon rpl 1 /L conssts of xactly on sgmnt, whch corrsponds to th sgmnt S of rpl/k as follows: a Th slop of rpl 1 /L s qual to th slop of S b Th assocatd nrtas of rpl 1 /L and S ar qual c For ach root δ of th assocatd polynomal A y of S th lmnt δ ps 1 of th assocatd polynomal of rpl 1 /L s a root Proof Th mnmal polynomal of α 1 ovr L 1 s p s 1 =1 x α So th slop of rpl 0 /L 1 s qual to th slop of S 1 By th Thorm of th Product [14, 3] th assocatd polynomal of rpl 0 /L 1 s A y Thus a, b, and c hold for = 1 Th clams a, b, and c follow by nducton on by Lmma 61 7 Splttng Flds In ordr to dtrmn th Galos group of an Esnstn polynomal ϕx w look at ts splttng fld If th ramfcaton polynomal of ϕx conssts of on sgmnt, w can xplctly construct th splttng fld of ϕx In th gnral cas w consdr th splttng flds of th subflds corrspondng to th sgmnts of th ramfcaton polygon and obtan th splttng fld of ϕx as a p-xtnson ovr thr compostum 11

12 Ramfcaton Polygons wth On Sgmnt Assum that th Nwton polygon of ρx s a straght ln It follows from Lmma 41 that ths can only b th cas f thr p n or n = p m for som postv ntgr m Snc w hav tratd th cas p n n scton 2, w assum n = p m Th tamly ramfd subfld of th splttng fld of ϕx s th splttng fld of th ramfcaton polynomal ρx of ϕx W frst consdr th splttng fld of such a polynomal ρx whos dgr s not dvsbl by p Lmma 71 Assum that th Nwton polygon of ρx O L [x] conssts of on sgmnt of slop h/ wth gcdh, = 1 = a + bh for a, b Z and gcd, p = 1 Assum that ts assocatd polynomal Ay L[y] s squar fr and lt f b ts assocatd nrta Lt I/L b th unramfd xtnson of dgr lcmf, [Lζ : L] and lt ε O I wth A ε = 0 Thn N = I εb π s th splttng fld of ρx Proof Dnot by Ax O L [x] a lft of Ay Lt M/K b th mnmal unramfd xtnson ovr whch Ay splts nto lnar factors, say Ay = y γ 1 y γ n/ ovr M Lt N = Mβ, ζ whr β s a root of ρx and ζ s an -th root of unty Lt γ = β /π h thn Aγ = 0 Th fld N s th splttng fld of ρx, f ρx or quvalntly ρβx, splts nto π hn/ lnar factors ovr N W obtan ρβx γπ h n/ x γ 1 γ x γ n/ mod π N O N [x], γ whr π N dnots a unformzr n th valuaton rng O N of N As gcd, p = 1 for 1 n/ th polynomals x γ ar squar fr ovr N Bcaus ζ γ N, thy splt nto lnar factors ovr N Hnsl lftng ylds a dcomposton of nto lnar factors It follows ρβx γπ h n/ that ρx splts nto lnar factors ovr N, thus N s th splttng fld of ρx Ovr M th polynomal ρx splts nto rrducbl factors θ x = j=0 θ,jx j 1 n/ whr θ,0 γ π h mod π h+1 By Proposton 21 th xtnsons gnratd by th θ x ar somorphc to th xtnsons gnratd by th polynomals x + γ π h b π a = x + γ b π wth a + bh = 1 Lmma 72 Lt u a powr of p Lt F x = r =0 a x p F u [x] b an addtv polynomal and assum N s a dvsor of u 1 and of all p 1 for all 1 r wth a 0 If 1 F u s a root of Gx = r =0 a x p 1/, thn F x splts nto lnar factors ovr F u f and only f Gx splts nto lnar factors ovr F u Proof by Ptr Müllr Clarly, f F x splts nto lnar factors thn Gx splts nto lnar factors, as th roots of Gx ar powrs of th roots of F x Lt E b th splttng fld of x 1 ovr F p Snc u 1 w hav E F u Lt M b th st f roots of F x n th algbrac closur of F p As F x s addtv M s addtvly closd Furthrmor, f λ E and v M thn λv M, bcaus w had assumd that E s a subfld of F p for all 1 r wth a 0 Hnc M s an E-vctor spac For ach 0 v M th lmnt v s a root of Gx and thrfor v s contand n F u So v u 1 = 1 and, as E contans th -th roots of unty, v u 1 E Thus thr xsts λ v E wth v u = λ v v 12

13 Now assum that v M but v E As G1 = 0 w hav 1 M It follows from λ v+1 v + 1 = v + 1 u = v u + 1 u = λ v v + 1 and th lnar ndpndnc of 1 and v, that 1 = λ v+1 = λ v Hnc M F u, so F x splts nto lnar factors ovr F u Thorm 73 Lt ϕx O K [x] b an Esnstn polynomal of dgr np m and assum that ts ramfcaton polygon rpϕ conssts of on sgmnt of slop h/ whr gcdh, = 1 = a + bh for a, b Z Lt α b a root of ϕx, L = Kα and Ay L[x] th assocatd polynomal of rpϕ wth assocatd nrta f Thn N = I εb α s th splttng fld of ϕx whr I/L s th unramfd xtnson of dgr lcmf, [Lζ : L] and ε K s arbtrary wth A ε = 0 Proof By th constructon of th ramfcaton polynomal ρx th splttng fld of ρx ovr L s th splttng fld of ϕx ovr K To b abl to us Lmma 71 to fnd th splttng fld of ρx, w nd to show that Ay s squar fr Lt ρx = n =0 ρ x O K [x] b th ramfcaton polynomal of ϕx Thn th assocatd polynomal to rpϕ s Ay = n 1/ j=0 A j y j = n 1/ j=0 ρ j α hj n 1/ y j L[y] W consdr th polynomal Bx = n =0 B x = xaγx for a root γ of Ay It follows from th constructon of Ay that A j 0 f th corrspondng coffcnt ρ j of ρx ylds a vrtx of rpϕ By Lmma 41 f B 0 thn = p s for som s {0,, m} Thus Bx s an addtv polynomal Furthrmor B x = B 1 = A 0, so gcdbx, B x = 1 and thrfor Bx and Ax ar squar fr It rmans to b shown that lcmf, [Lζ : L] s th dgr of th splttng fld of Ay ovr F q = K Bcaus th assocat nrta f s th dgr of th splttng fld of Ay ovr F q and as q f 1 ths follow from Lmma 72 wth u = q f, F x = Bx and Gx := Aγx Th Gnral Cas In th gnral cas, that s whn rpϕ conssts of mor than on sgmnt, ramfcaton polygon and assocatd polynomals do not provd nough nformaton to dscrb th splttng fld compltly But w can us th rsults for on sgmnt and th corrspondnc of thorm 62 to gv a subfld T, such that th splttng fld of ϕx s a p-xtnson ovr T In othr words, th fld T contans th maxmal subfld of th splttng fld, whch has dgr coprm to p ovr th ground fld Thorm 74 Lt ϕx = x n + n 1 =0 ϕ x O K [x] b Esnstn of dgr n = p m wth p and m > 0 Assum th ramfcaton polygon rpϕ of ϕx conssts of l + 1 sgmnts S 1,, S l+1 For 1 l lt m = h / b th slop of S wth gcdh, = 1 = d + b h for d, b Z, A y O K [y] b th assocatd polynomal and f assocatd nrta of S, γ K such that A γ = 0, and v = b p m s 1 + n

14 Morovr w dnot by I th unramfd xtnson of K of dgr f = lcmf 1,, f l+1, [Kζ 1 : K],, [Kζ l : K] and by N th splttng fld of ϕx Lt α b a root of ϕx and Kα = L 0 L 1 L l K as n Thorm 62 b th towr of subflds corrspondng to rpϕ Thn: a Th fld T = I 1 γ n 1 ϕ 0,, l γ n l ϕ 0, ϕ 0 for 1 l s a subfld of N/K, such that N/T s a p-xtnson b For 1 l 1 th xtnsons TL 1 /TL ar lmntary Ablan c Th xtnson T/K s Galos and tamly ramfd wth ramfcaton ndx 0 lcm 1,, l Furthrmor [T : K] < n 2 Proof Assum th that th roots α = α 1,, α n of ϕx ar ordrd as n Lmma 52 For 1 l w hav L = Kβ wth β = α 1 α p Th conjugats of β ar of th form β j = α j 1p s +1 α jp s Thorm 73 ylds th normal closur N of L 1 /L By Thorm 62 w can us, b, and f of th sgmnt S whn dtrmnng N If ε s a root of A y thn, by Thorm 62 c, w gt N = I ε ps 1 b β 1 wth I /L 1 unramfd of dgr lcmf, [L 1 ζ, L 1 ] and δ O I a lft of a root of th assocatd polynomal to rpl 1 /L Furthrmor N /L s normal By Lmma 81 th frst ramfcaton group and thrfor th wldly ramfd part of N /L s lmntary Ablan For th tamly ramfd xtnson L l /K w st N l+1 = I l+1 = L l ζ 0 W now collct all unramfd xtnsons ovr K and consdr th towr of xtnsons IL IL 1 IL l I K By th dfnton of I th xtnsons IN /IL ar Galos an totally ramfd and thr tamly ramfd part IN /IL 1 s gnratd by x + 1 b ε b p s 1 β 1 Smlarly to th unramfd parts w now consdr th tamly ramfd parts ovr I Th mnmal polynomal of ε ps 1 b β 1 ovr I s th norm of ts mnmal polynomal of N /I : N IL 1 /I x + ε ps 1 b β 1 and ts constant trm s 1 b ε b p s [IL 1 :I] 1 1 n ϕ 0 Th product of th conjugats of β 1 s up to sgn qual to n =1 α = ±ϕ 0 So T = I 1 0 v ε b p n ϕ 0 s Galos and th tamly ramfd part of IN /I 1 l Each of ths xtnsons contans I/L l /I Th compostum of th T s T In th nw towr of xtnsons TL = TL 0 TL 1 TL l 1 T I K 14

15 th xtnson T/K s Galos, bcaus t s th compostum of Galos xtnsons Also TL 1 /TL s an lmntary Ablan p-extnson whch proofs b It follows by nducton that N/T s a p-xtnson Th ramfcaton ndx of T/K follows from Abhyankar s Lmma s, for xampl,??chaptr 5 2]nark, as T s th compostum of th tamly ramfd xtnsons T /K A frst, obvous, bound for [T : K] s 0 [Kζ 0 : K] n wth n f [Kζ : K] By Thorm 62 w can us th xtnson L 1 /L to stmat n for 1 l W obtan l n < p s 1 p s 2 s1 p s l s l 1 2 = p s l 2 = p m 2 =1 Furthrmor 0 [Kζ 0 : K] < 2 0 whch mpls c 8 Galos Groups In th cas of an Esnstn polynomal ϕx wth on-sdd ramfcaton polygon w us th rsults of scton 7 and th wll known structur of Galos groups of tamly ramfd xtnsons thorm 23 to gv an xplct dscrpton of Galϕ Rcall, that w dnot th slop of th ramfcaton polygon by h/ gcdh, = 1 and that th dgr of ϕx s qual to p m Dnot by N th splttng fld of ϕx, by L th subfld gnratd by a root of ϕx, and by T th maxmal tamly ramfd subfld of N/K By Lmma 71 T/K has ramfcaton ndx and ts nrta dgr f s dtrmnd by th dgrs of th rrducbl factors of th assocatd polynomal ovr K St G = Galϕ = GalN/K, H = GalN/L, and lt G 1 G b th frst ramfcaton subgroup of N/K Thn G = G 1 H holds, as L and T satsfy th condtons L T = K and LT = N Bcaus H s th Galos group of a tam xtnson, ts structur s wll known Thorm 23 It rmans to dtrmn th group G 1 and th acton of H on G 1 W dnot by G th -th ramfcaton subgroup of G In th followng, w xamn th ramfcaton fltraton G G 0 G 1 of G Lmma 81 Th ramfcaton fltraton of G = Galϕ s G G 0 G 1 = G 2 = = G h > G h+1 = {d} Th group G 1 = GalN/K s somorphc to th addtv group of F p m Proof Lt π N b a prm lmnt of N W hav to show th qualty ν N π g N π N = h + 1 for all g G 1 As N = LT th ramfcaton polygon rpn/t s a ln of slop h = h Thus ν N πg N π N π N = h for all g G and thrfor w obtan ν N π g N π N = ν N π g N π N π N + ν N π N = h + 1 for all g G 1 as dsrd Snc th quotnts G /G +1 for 1 mbd nto th addtv Group of th rsdu class fld of N s [19, chaptr IV], th scond statmnt follows from G 1 = G h = G h /G h+1 Th nxt thorm spcfs th acton of H on G 1 and dscrbs th Galos group G as a subgroup of th affn group AGLm, p W dnot by = π N th maxmal dal of 15

16 th valuaton rng O N Th group G acts naturally on th quotnts / +1 whch ar, as addtv groups, somorphc to th addtv group of th rsdu class fld of N Furthrmor, π g Θ : G /G +1 / +1 N, + : gg π N mbds ach quotnt G /G +1 nto / +1 s agan [19, chaptr IV] Thorm 82 Lt ϕx O K [x] b an Esnstn polynomal of dgr p m, whos ramfcaton polygon conssts of on sngl sgmnt of slop h wth gcdh, = 1 Thn Galϕ = G 1 H, whr G 1 s th frst ramfcaton group and H corrsponds to th maxmal tamly ramfd subfld of th splttng fld of ϕx s Proposton 73 Morovr, Galϕ s somorphc to th group G = {t a,v : F p m F p m : x xa + v a H GLm, p, v F p m } of prmutatons of th vctor spac F p m, whr H dscrbs th acton of H on Θ h G h /G h+1 h / h+1 s dfnton abov Proof W hav alrady sn, that Galϕ = G 1 H If G 1 = {s v : x x + v v F p m } and H = {u a : x xa a H } thn G = G 1 H, whr th acton of H on G1 s th multplcaton of a vctor by a matrx: s ua v : x xa 1 + va = x + va Frst of all, w hav G 1 = G1 = G h /G h+1 by Lmma 81 Now, w rlat th actons of H on h / h+1 and on G 1 Th njctv homomorphsm π g Θ h : G h /G h+1 = G 1 h / h+1 N : g 1 mod h+1 π N s a H-homomorphsm, whch mans, that Θ h g b = Θ h g b for g G 1, b H To s that, lt π g N = π N1 + δ wth δ h Thn Θ h g b = δ b mod h+1 For computng Θ h g b = π gb N π N 1 mod h+1, st πn b 1 = π N ε wth ε O N and consdr πgb N = gb πb 1 N = π N ε gb = π g N εg b Ths s modulo h+1 congrunt to π g N εb = π N 1+δε b = π N ε b 1+δ b = π N 1+δ b whch provs th assrton It follows, that H acts on G 1 n th sam way as t acts on Θ h G 1 h / h+1, whr both groups ar somorphc to F p m, + Bcaus th acton on Θ h G 1 must b fathful, th acton on G 1 s fathful, too Lt H b th subgroup of GLm, p, whch dscrbs th acton of H on Θ h G 1 Thn H = H = H and thus Galϕ = G Rmark 83 Anothr way to dscrb th Galos group s as a fntly prsntd group: Galϕ s = s, t, a 1,, a = 1, t f = s r, s t = s q, [a, a j ] = 1, m a p = 1, as = s, a t = t for 1 < j m Hr s and t gnrat a subgroup somorphc to H and a 1,, a m gnrat a normal subgroup somorphc to G 1 Th numbr s th dnomnator of th slop of rpϕ and f s qual to lcmf 1, [Kζ : K], whr f 1 dnots th assocatd nrta Th ntgr r fulflls th condton Aζ r = 0 for a prmtv q f 1-th root of unty ζ and th assocatd polynomal Ay L[y] compar lmma 71 and thorm 23 Th lmnts s and t ar words n a 1,, a m Thy ar dtrmnd by th acton of th gnratng automorphsms of H on Θ h G 1 = Θ h G h /G h+1 h / h+1 16

17 In both dscrptons of Galϕ w nd a lttl computaton to gt th acton of H on Θ h G 1 and thrfor th matrx group H or th lmnts s and t As w hav two gnrators of H xplctly as automorphsms of N/L s 23, w can dtrmn thr acton on h / h+1 W us th rprsntaton of H of dmnson f f ovr F p, whr f dnots th nrta dgr of th ground fld K Now, w hav to fnd th submodul Θ h G 1 = F + p of th H-modul m h / h+1, + = F +, q f as, n gnral, q f p m In th followng lmma w show that Θ h G 1 can asly b computd form th zros of th assocatd polynomal Ay K[y] Rcall that th splttng fld N of ϕx O K [x] can b rprsntd n th form N = Lζ, π N = Kαζ, π N wth π N = ζ r α and lt d = pm 1 b th dgr of th assocatd polynomal Ay Lmma 84 Lt γ 1,, γ d b th zros Ay n N and a, b N wth a bp m = 1 Thn: a For 1 d th rsdu class fld N contans th -th roots of by γ,1,, γ, b Th mags of G 1 undr Θ h ar {0 + h+1, aγ,j π h N + h+1 1 d, 1 j }, whr γ,j dnots a lft of γ,j N to O N γ ζ rh whch w dnot Proof a Th roots 1 + α s α pm of th ramfcaton polynomal ρx hav N- valuaton h and thrfor th form ξπn h for som ξ O N By Lmma 31 th roots of Ay ar of th form ξπ h N ξ ζ r α = = ξ ζ rh α h b Th homomorphsm Θ h : G h /G h+1 h / h + 1 s ndpndnt of th choc of th prm lmnt As n th proof of Lmma 44 w thrfor can us th prm lmnt π N = αa /β b, whr β s an unformzng lmnt of T Also as n th proof of Lmma 44, w us th rprsntaton α /α = 1 + δα h/ wth νδ = 0 for th roots of ρx Not that δ and α h/ n gnral ar not lmnts of N Now w hav for σ G 1, bcaus p a, that σπ N π N α h 1 = αa β b β b α 1 = α a α a 1 = αδα h/ + Th mag Θ h σ s th cost of aδα h/ + n h /wp h+1 In ordr to fnd a rprsntatv by lmnts n N for ths class w start wth ξ ζ r α h = δα h/ whch s quvalnt to δ = γ ζ rh Thus, as ξ O N : σπ N 1 = aξ ζ π N rh α h + = aξπn h + aξπn h mod h+1 Snc ach of th d = p m 1/ roots of Ay gv lmnts ξ, w hav, togthr wth 0 + h+1, dscrbd all mags of Θ h Rmark 85 W us th notaton from th proof of Lmma 84 abov Th opraton of th fld automorphsms s and t s Thorm 23 on Θ h G 1 ar sζ π h N + h+1 = ζ lh+ π h N + h+1 and tζ π h N + h+1 = ζ lhk+q π h N + h+1 17

18 9 Exampls W gv som xampls to dmonstrat th calculaton of Galos groups usng our rsults W consdr two polynomals of dgr 9 ovr Q 3 ladng to dffrnt Galos groups and on polynomal of dgr 81 ovr Q 3 In ach of th xampls w dnot by L = Kα th fld gnratd by a root α of th rspctv polynomal Exampl 91 romano xampl51 Exampl 92 W dtrmn th Galos group of ϕx = x 9 + 9x + 3 Q 3 [x] Th ramfcaton polygon of ϕx s a straght ln connctng th ponts 0, 10 and 8, 0 of slop h = 5 Thrfor th polynomal ϕx s not covrd by Romano s rsults Lt α b 4 a root of ϕx and L = Q 3 α Th assocatd polynomal Ay = y L[x] = F 3 [x] of th ramfcaton polynomal ρx L[x] s rrducbl, so ts assocatd nrta s f = 2 Th nrta dgr of th splttng fld of ρx s lcm2, 4 = 4 Lt ζ b a prmtv ghth root of unty Bcaus ζ 2 s a root of Ay = y 2 + 1, Lmma 71 gvs us N = Lζ, 4 ζ 2 α as th splttng fld of ρx ovr L, whch s also th splttng fld of ϕx ovr Q 3 s Proposton 73 St π N = 4 ζ 2 α By Thorm 23 wth = 4, f = 2 and r = 2 th group H = GalN/L s gnratd by th automorphsms s : ζ ζ, π N ζ 2 π N and t : ζ ζ 3, π N ζπ N Wth Rmark 85 w gt S GL2, 3 as th rprsntaton matrx of th automorphsm of F + 3 gvn by ζ ζ 10+ and T GL2, 3 as th rprsntaton matrx of th automorphsm 2 ζ ζ 3 Wth a bass corrspondng to 1, ζ of 5 / 6 = F3 2, +, w obtan S = and T = whch rprsnt th acton of s and t on th frst ramfcaton group G 1 Th matrcs S and T gnrat a rprsntaton of th quatrnon group Q 8 of ordr 8 ovr F 3 In ths spcal cas w ar alrady n th rght dmnson 2 and t s not ncssary to sarch for a submodul Hnc th Galos group of ϕx s somorphc to th group G = { t a,v : F 3 2 F 3 2 : x xa + v a S, T, v F3 2} = C 2 3 Q 8 Exampl 93 Lt ϕx = x 9 + 3x Q 3 [x] Hr, th ramfcaton polygon conncts th ponts 0, 2 and 8, 0, thrfor has slop h = 1 Th assocatd polynomal of 4 th ramfcaton polynomal ρx L[x] s congrunt to x n L[x] As x splts nto lnar factors ovr L = F 3, th polynomal ρx gnrats totally and tamly ramfd xtnsons of dgr 4 of L Hnc w must add th 4-th roots of unty to gt th splttng fld N = Lζ, 4 α s Lmma 71 and Proposton 73, whr ζ s a prmtv ghth root of unty By Thorm 23 wth = 4, f = 2 and r = 0 th group H = GalN/L s gnratd by th automorphsms s : ζ ζ, 4 α ζ 2 4 α and t : ζ ζ 3, 4 α 4 α 18

19 Wth a bass corrspondng to 1, ζ of / 2 = F3 2, +, w obtan S = and T = for th acton of s and t on G 1 Agan, w ar alrady n th rght dmnson and do not hav to sarch for a submodul In ths cas S and T gnrat a rprsntaton of th dhdral group D 8 of ordr 8 ovr F 3 and Galϕ s somorphc to G = { t a,v : F 3 2 F 3 2 : x xa + v a S, T, v F3 2} = C 2 3 D 8 Exampl 94 W dtrmn th Galos group of th polynomal ϕx = x x x x x Q 3 [x] Lt α b a root of ϕx and L = Q 3 α Th ramfcaton polygon of ϕx s a straght ln connctng th ponts 0, 10 and 80, 0 of slop h = 1 Th assocatd polynomal of 8 th ramfcaton polynomal ρx L[x] s Ay = y = y + 1y + 2y 4 + y 4 + L[x] Hnc th assocatd nrta s 4 Wth [Q 3 ζ 8 : Q 3 ] = 2 w obtan th nrta dgr f = lcm4, 2 = 4 of th splttng fld So T = F 3 4 Lt ζ b a th root of unty Bcaus ζ 0 = 1 s a root of Ay = y , Lmma 71 gvs us N = Lζ, 8 α as th splttng fld of ρx ovr L, whch s also th splttng fld of ϕx ovr Q 3 s Proposton 73 By Thorm 23 wth = 8, f = 4 and r = 0 th group H = GalN/L s gnratd by th automorphsms s : ζ ζ, 8 α ζ 10 8 α and t : ζ ζ 3, 8 α 8 α By Rmark 85 w obtan S GL4, 3 as th rprsntaton matrx of th automorphsm of F + 3 gvn by ζ ζ 10+ and T GL4, 3 as th rprsntaton matrx of th automorphsm 4 ζ ζ 3 Wth a bass corrspondng to 1, ζ, ζ 2, ζ 3 of / 2 = F3 4, +, ths ar S = and T = Hnc th Galos group of ϕx s somorphc to th group of ordr = 2592 G = { t a,v : F 3 4 F 3 4 : x xa + v a S, T, v F3 4} = C 4 3 S, T 10 Acknowldgmnts W thank Ptr Müllr from Unvrstät Würzburg for th proof of Lmma 72 Support was provdd n part by a nw faculty grant from UNC Grnsboro and by a grant from th Dutsch Forschungsgmnschaft 19

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