CLASSIFYING EXTENSIONS OF THE FIELD OF FORMAL LAURENT SERIES OVER F p

CLASSIFYING EXTENSIONS OF THE FIELD OF FORMAL LAURENT SERIES OVER F p JIM BROWN, ALFEEN HASMANI, LINDSEY HILTNER, ANGELA RAFT, DANIEL SCOFIELD, AND IRSTI WASH Abstract. In prvious works, Jons-Robrts and Pauli-Roblot hav studid finit xtnsions of th p-adic numbrs Q p. This papr focuss on rsults for local filds of charactristic p. In particular w ar abl to produc analogous rsults to Jons-Robrts in th cas that th charactristic dos not divid th dgr of th fild xtnsion. Also in this cas, following from th work of Pauli-Roblot, w prov that th dfining polynomials of ths xtnsions can b writtn in a simpl form amnabl to computation. Finally, if p is th dgr of th xtnsion, w show thr ar infinitly many xtnsions of this dgr and thus ths cannot b classifid in th sam mannr. 1. Introduction Classifying xtnsions of Q p has bn of intrst for many yars. Pauli and Roblot [11] dscrib a mthod for computing dfining polynomials for all xtnsions of Q p of a givn dgr. Jons and Robrts [9] constructd an onlin databas that idntifis dgr n xtnsions of Q p for small valus of p and n. Thy dscrib how to comput various invariants for ach xtnsion, including th Galois group. In a similar fashion, w xtnd ths rsults to charactristic p local filds, focusing on th unramifid, totally tamly ramifid, and totally wildly ramifid cass. W bgin by introducing th radr to ssntial background topics. Givn a charactristic p local fild F and an intgr n rlativly prim to p w classify all dgr n xtnsions of F. W rcall th rsult that for ach f n thr is a uniqu unramifid xtnsion of dgr f. W thn turn our attntion to totally tamly ramifid xtnsions of dgr = n/f. W follow th work of Jons and Robrts [9] to comput a class of dfining polynomials for ths xtnsions, namly a spcific typ of Eisnstin polynomial. W nxt considr th totally wildly ramifid cas whn n = p. Our rsults for dgr p xtnsions ar not analogous to th cas of charactristic zro local filds, as thr ar infinitly many dgr p xtnsions. W conclud by classifying all dgr 10 xtnsions of F p ((T )) whr p ±3 (mod 10). In particular, in th cas that p = 3 w giv spcific dfining polynomials for ach xtnsion. This illustrats computationally how on handls a spcific dgr and charactristic. Dat: Dcmbr 18, 2012. Th authors would lik to thank vin Jams, Rodny aton, Mohammd Tsmma, and Dania Zantout for hlpful convrsations during th REU. Thy would also lik to thank th rfr for valuabl suggstions that improvd th xposition of this papr. Th authors wr supportd by NSF DMS-1156734. 1

2. Background 2.1. Local Filds. This papr will b concrnd with xtnsions of local filds. W rfr th radr to [6, 14] for mor dtails on local filds. Lt F b a local fild. Lt π F dnot a uniformizr of F and writ O F for th valuation ring of F, M F = (π F ) for th maximal idal, and rsidu fild O F /M F. W normaliz th valuation on F so that ν F (π F ) = 1. Throughout this papr L/F will always rfr to a finit xtnsion of local filds. Givn L/F of dgr n on has π F = πl for som intgr 1 with n. W call th ramification indx of L/F and f = n/ th inrtia dgr of L/F. W say L/F is unramifid if = 1 and totally ramifid if = n. If O F /M F has charactristic p, w say L/F is tamly ramifid if p and wildly ramifid if p. On knows that th compositum of unramifid xtnsions is again unramifid, so on can form a maximal unramifid xtnsion F ur of F. Givn an xtnsion L/F of local filds, w st = L F ur. Clarly is th maximal unramifid xtnsion of F in L. Not th xtnsion L/ is ncssarily totally ramifid. 2.2. Th Fild of Formal Laurnt Sris. W will b intrstd in finit xtnsions of th fild of formal Laurnt sris. W now introduc this fild. Lt F p [T ] b th polynomial ring with cofficints in F p and F p (T ) its fraction fild. Dfinition 2.1. Givn x F p (T ), writ x as T r g with g, h F h p[t ], T gh. W dfin a valuation ν T by: ( ν T T r g ) = r h with ν T (0) =. Not that w can dfin ν f for any irrducibl polynomial in F p [T ] analogously. As th valuations arising in this mannr ar non-archimdan, thy giv th charactristic p valuations analogous to th p-adic valuations on Q. Morovr, on can dfin a valuation with rspct to 1/T to obtain th charactristic p valuation analogous to th usual absolut valu on Q. As w will only b intrstd in th cas f = T, w rstrict to that cas. W can now complt F p (T ) with rspct to ν T to obtain th fild of formal Laurnt sris ovr F p. Dfinition 2.2. A formal Laurnt sris f(t ) is an infinit sris of th form a i T i i= m with m, i Z, a i F p for all i. W dnot th st of such sris by F p ((T )). An quivalnt xprssion for th valuation dfind abov is ν T (x) = ν T ( i= m a i T i ) = m. W also dfin an absolut valu T such that T r g h T = p r. Not that F p ((T )) is a non-archimdan local fild with charactristic p. As w will only discuss th valuation on F p ((T )), w will b using th notation ν(x) rathr than ν T (x) to 2

dnot this spcific valuation for th rmaindr of th papr unlss othrwis spcifid. Givn an xtnsion L/F p ((T )) w dnot th valuation on L obtaind by xtnding ν by ν L. For th rst of th papr all our filds will b xtnsions of F p ((T )) for som prim p. In particular, F will b fixd to b a finit xtnsion of F p ((T )). 2.3. Ramification Groups. Lt L/F b a Galois xtnsion of local filds with Galois group G. W dfin th ramification groups of L/F by G i = {σ G : ν L (σ(x) x) i + 1 for all x O L } whr i 1. Th ramification groups mak up a chain of subgroups of th Galois group that ar vntually trivial. Ths G i may not b distinct for all i. Dfinition 2.3. In th subgroup chain of ramification groups, a ramification brak is dfind to occur at i 0 such that G i G i+1. Dpnding on th Galois group and ramification groups thmslvs, this brak may b uniqu. Not that th chain of ramification groups is an invariant of th fild, so distinct chains giv distinct filds. 3. Unramifid Extnsions Unramifid xtnsions of charactristic p filds ar similar to thir charactristic zro countrparts. W hav th following thorm in this rgard. Thorm 3.1. [8, p. 167] Lt F b a local fild and f b a positiv intgr. Thn F has a uniqu unramifid xtnsion of dgr f. This xtnsion is obtaind by adjoining a primitiv (p f 1)st root of unity to F. In particular, w s that if w wish to classify xtnsions of dgr n of a local fild F, it is nough to classify all th totally ramifid xtnsions of dgr for ach n. 4. Totally Ramifid Extnsions As notd in th prvious sction, unramifid xtnsions ar alrady wll undrstood. Thus whn w build up our dgr n xtnsion of F w focus on building totally ramifid xtnsions of dgr for ach n. Dfinition 4.1. Lt g(x) O F [x] b a monic polynomial: g(x) = x + a 1 x 1 +... + a 0. If ν(a i ) 1 for ach i = 0,..., 1, and ν(a 0 ) = 1, thn g(x) is said to b Eisnstin. Th following is a wll-known thorm which dscribs how to construct totally ramifid xtnsions. Thorm 4.2. [6, p. 54] A finit xtnsion L/ of a non-archimdan local fild is totally ramifid if and only if L = [α], with α a root of an Eisnstin polynomial. 3

4.1. Totally Tamly Ramifid Extnsions. Using th work of Pauli and Roblot [11], w can show xactly what th totally tamly ramifid xtnsions look lik, but first w nd som thorms adaptd from Pauli [12]. W lt /F b an unramifid xtnsion of dgr f and considr totally tamly ramifid xtnsions L/ of dgr. W dfin M := π. Dfinition 4.3. Lt L/ b a dgr Galois xtnsion with Galois group G. Lt (δ 1,, δ ) b an intgral basis of L/. Writ G = {σ 1,..., σ }. Thn is th discriminant of L/. disc(l/) = (dt(σ l (δ k )) 1 k,1 l ) 2 Th discriminant of th fild gnratd by an Eisnstin polynomial is xactly th discriminant of th polynomial. Lmma 4.4. Lt L = (α)/ b a finit Galois xtnsion of dgr with basis lmnts 1, x, x 2,, x 1 and g b th minimal polynomial ovr with roots α 1,..., α whr α = α 1. Thn disc(l/) = disc(g) and ν (disc(g)) = ν (g (α)). Proof. Dfin σ i Gal(L/) such that σ i (α) = α i for i {1,..., }. Thn σ i (x j ) = α j i for 0 j 1. Not disc(l/) is th squar of th dtrminant of th matrix 1 x 1... x 1 1 1 x 2... x 1 2 A =... 1 x... x 1 Sinc A is a Vandrmond matrix, dt A = i<j (α i α j ) and it follows that disc(l/) = disc(g). On th othr hand, for any y L w can writ g(y) = (y α 1 ) (y α n ), so w hav g (α i ) = (α i α j ). k j k Howvr, only th k = i trm is non-zro. Hnc g (α i ) = j i(α i α j ),. so it follows that Thrfor, disc(g) = ν (disc(g)) = ν ( g (α i ). i=1 g (α i )) = ν (g (α i )). Lmma 4.5. If x 0,, x 1 whr x i x j for i j, thn 1 x i = max 0 i 1 { x i }. i=0 i=1 4

Thorm 4.6. (Or s Conditions) For ach n thr xists a totally ramifid xtnsion L/ of dgr and discriminant M 1. Proof. By Thorm 4.2, vry totally ramifid xtnsion L of of dgr can b gnratd by adjoining a root α of an Eisnstin polynomial g(x) = x + a 1 x 1 +... + a 0. W hav disc(l/) = disc(g(x)) and sinc g(x) is Eisnstin, w can writ ν (disc(g(x)))/ = ν (g (α)) bcaus g(x) is irrducibl. Sinc α is a uniformizr in L, ν (α) = 1/. Th valuations of ia i α i 1 for 1 i < and α 1 ar all diffrnt and so by Lmma 4.5 w gt ν (g (α)) = ν (α 1 + ( 1)a 1 α 1 +... + a 1 ) { = min 1 i 1 ν () + 1, ν (i) + ν (a i ) + i 1 Not that ν (x) = 0 for all x Z and ν (a i ) 1 for all 1 i 1, so { 1 ν (g (α)) = min, ν (a i ) + i 1 } 1 i 1 = 1. Thus sinc g(x) is irrducibl and ν (disc(g(x))) = ν (g (α)) = 1 it is clar that w can construct an Eisnstin polynomial g(x) such that disc(g(x)) = M 1. 4.2. Construction of Gnrating Polynomials. Lt L dnot th st of all totally ramifid xtnsions L/ of dgr and discriminant M 1. In this sction w us th work of [11,12] to construct a finit st of polynomials that will gnrat all th xtnsions in L. As abov, w lt /F b an unramifid xtnsion of dgr f and L/ b a totally ramifid xtnsion of dgr. Lt H b th Galois group of th xtnsion /F and lt R 1,2 b a fixd H-stabl systm of rprsntativs of th quotint M 1 /M2. W dnot R 1,2 to b th subst of R 1,2 whos lmnts hav ν -valuation 1. Lt Ω b th st of -tupls (ω 0,..., ω 1 ) which satisfy th following conditions: { R (1) ω i 1,2 if i = 0, R 1,2 if 1 i 1. }. To ach lmnt ω = (ω 0,..., ω 1 ) Ω w associat th polynomial A ω (x) O [x] givn by A ω (x) = x + ω 1 x 1 + + ω 1 x + ω 0. Lmma 4.7. Th polynomials A ω ar Eisnstin polynomials of discriminant M 1. Proof. By construction ν (ω i ) 1 for all i and ν (ω 0 ) = 1. So A ω is an Eisnstin polynomial. Lt α b a root of A ω. Sinc th discriminant of A ω is th norm from (α) to of A w(α) w hav ν (A w(α)) = 1 as sn in Thorm 4.6. It follows that ν (disc(a ω )) = 1 and disc(a ω ) = M 1 as claimd. 5

Lmma 4.8. Lt ω b an lmnt of Ω and lt α b a root of A ω (x). Th xtnsion (α)/ is a totally ramifid xtnsion of dgr and discriminant M 1. Convrsly, if L/ is totally ramifid xtnsion of dgr and discriminant M 1, thn thr xists ω Ω and a root α of A ω (x) such that L = (α). Proof. Th statmnt is a spcial cas of th charactristic zro rsult Corollary 5.3 in [11]. In particular, on spcializs to j = 0 and c = 2. Th proof thr works for charactristic p as wll. Thorm 4.9. Lt q b th ordr of th rsidu fild of. Thn th numbr of totally ramifid xtnsions of of dgr and discriminant M 1 is #L =. Proof. To s this, on combins Lmma 6.2 and Lmma 6.3 of [11] and obsrvs th proofs carry ovr vrbatim to charactristic p. Pauli and Roblot hav calculatd convnint polynomials that gnrat totally tamly ramifid xtnsions of unramifid xtnsions of Q p. Thir proof carris ovr to th positiv charactristic cas as wll. W includ th proof for th convninc of th radr. Thorm 4.10. Lt ζ b a primitiv (p f 1)-st root of unity containd in and lt g = gcd(p f 1, ). St m = /g. Thr ar xactly totally and tamly ramifid xtnsions of of dgr. Furthrmor, ths xtnsions can b split into g classs of m -isomorphic xtnsions, all xtnsions in th sam class bing gnratd ovr by th roots of th polynomials for r = 0,..., g 1. f r (x) = x ζ r π Proof. Considr th st R 1,2 = {ζ i π with 0 i p f 2} and R 1,2 = R 1,2 {0}. Th roots of th polynomials x + ω 1 x 1 +... + ω 0, whr ω i R 1,2 for 1 i 1 and ω 0 R 1,2, gnrat all totally tamly ramifid xtnsions of discriminant M 1 by Lmma 4.8. Considr xtnsions of gnratd by roots of th polynomials f i (x) = x ζ i π so that ω j = 0 for 1 j 1. Lt α b a root of f i (x). Not that sinc ζ, w hav ζ h α gnrats th sam xtnsion of as α for any intgr h. If w choos h so that h + i r (mod p f 1) with 0 r < g, thn th minimal polynomial of ζ h α is f h+i (x) sinc (ζ h α) + ζ h+i π = ζ h α + ζ h+i π = ζ h (α + ζ i π ). Hnc w only nd to considr th polynomials f r (x) for 0 r g 1. This polynomial is Eisnstin and by Thorm 4.2, it will dfin a totally tamly ramifid xtnsion. Lt f r (x) and f r (x) b two of ths polynomials which gnrat a totally tamly ramifid xtnsion whr 0 r, r g 1 and r r. Lt α and α b roots of f r (x) and f r (x) rspctivly. Suppos that α and α gnrat th sam fild L. Thn this fild contains an -th root of ζ r r. To s this, considr th following: If w assum α L if and only if 6

α L thn f r (α) = 0 = f r (α ). Thus α (α ) = ζ r π ζ r π = π (ζ r ζ r ) = ζ r π (1 ζ r r ). Thus this fild contains an -th root of ζ r r which contradicts our assumption that th fild only contains th (p f 1)-st roots of unity as r r is nvr a multipl of modulo p f 1. Thrfor α and α must gnrat two distinct xtnsions of. Lt ρ b a primitiv -th root of unity in th algbraic closur of F p ((T )) such that for m, ρ m = ζ (pf 1)/g. Th conjugats of α ovr ar α, ρα,..., ρ 1 α. Thus α, ρ m α,..., ρ (g 1)m α all gnrat th sam fild, but α, ρα,...ρ m 1 α all gnrat distinct isomorphic xtnsions. Mor spcifically, th roots of th polynomial f r (x) gnrat g classs of m distinct isomorphic xtnsions. Thus thr ar total xtnsions gnratd by th roots of ths polynomials. By Thorm 4.9 thr ar xactly totally ramifid xtnsions of dgr of, which provs that all totally tamly ramifid xtnsions of dgr of ar gnratd by th roots of th polynomials f r (x) as claimd. Thus, w hav shown that th polynomials calculatd in [11] to gnrat totally tamly ramifid xtnsions of of dgr whr p also work in th cas of char() = p. 4.3. Totally Wildly Ramifid Extnsions of Dgr p. In this sction w discuss wildly ramifid xtnsions L/F of dgr p. W show that th charactristic p thory diffrs significantly from charactristic zro thory and thus it is not possibl to classify such xtnsions as in som charactristic zro cass [2 4]. Artin-Schrir thory provids th rsults ndd for ths xtnsions. From this thory, th Galois group G = Gal(L/F ) will b cyclic, namly Z/pZ. Bcaus of that fact, th ramification groups will ithr b G or {1} causing thr to b a singl, uniqu ramification brak. For mor on Artin-Schrir thory, s [6, p. 67-78]. Not also that in this sction, th group U i, which corrsponds to th ramification group G i, will b writtn as ithr 1 + (π i L ) or 1 + Mi L. Dfinition 4.11. For F a fild of charactristic p, an Artin-Schrir polynomial is a polynomial of th form (x) = x p x α for α F. Th following is a wll-known rsult that lads to our nxt thorm. Lmma 4.12 (Hilbrt s Thorm 90, Additiv Form). Lt L/F b a cyclic Galois xtnsion with dgr n and Galois group G. Lt σ b a gnrator of G and lt β L. Thn Tr L/F (β) is qual to 0 if and only if thr xists α F such that β = α σ(α). Thorm 4.13. [10, p. 290] Any Galois xtnsion of F of dgr p is th splitting fild of an Artin-Schrir polynomial. Proof. Lt L/F b a Galois xtnsion of dgr p. Thn Tr L/F ( 1) = p( 1) = 0 sinc F has charactristic p. Lt σ b a gnrator of G. By Hilbrt s Thorm 90 thr xists α L such that σ(α) α = 1. Thus σ(α) = α + 1 and σ i (α) = α + i for i = 1,..., p. Sinc α has p distinct conjugats, [F (α) : F ] p. It follows that L = F (α). Not that σ(α p α) = σ(α) p σ(α) = (α + 1) p (α + 1) = α p α. 7

Sinc α p α is fixd by σ, th gnrator of G, it is fixd by vry lmnt of G. Hnc α p α F. Lt a = α p α. Thn α satisfis th quation x p x a = 0 and L/F is th splitting fild of an Artin-Schrir polynomial. Thorm 4.14. Thr ar infinitly many wildly ramifid xtnsions of dgr p of F. Proof. Lt L b th splitting fild of th polynomial (x) = x p x π m F F [x] with m Z. Suppos L/F is a wildly ramifid xtnsion of dgr p with ν L a discrt valuation on L and G th Galois group. Lt L b a uniformizr. It suffics to show that thr ar an infinit numbr of valus at which th ( uniqu ramification ) brak can occur. Considr ν L (σ( ) ) = 1 + ν σ(πl ) L 1. With this quality, in G i w can look ) ( at ν σ(x) L 1 i rathr than ν x L (σ(x) x) i + 1. It can b found in [14, p. 67] that σ() U L. Thus, σ() = u for som unit u U L. Lt u = u F w for u F U F and w 1 + M L. Thn w hav, ( ) σ(πl ) σ σ() = σ(u F w) u F w = u 2 F w σ(w). Continu this procss of multiplying by σ() = σ(u F w) on ach sid until, on th lft hand sid, th trm is qual to σp ( ). Bcaus this is a dgr p xtnsion with cyclic Galois group, 1 = σp ( ) = u p F wσ(w) σp 1 (w) whr wσ(w) σ p 1 (w) 1 + M L. Divid by wσ(w) σ p 1 (w) to s u p F 1+M L. This implis u F 1+M L and u F 1+M F. Thn, σ() 1 + M L. This givs σ() = 1 + u L πl s for som u L U L and s 1, whr s dos not dpnd of choic of uniformizr. From [14, p. 66-67], σ(u) 1 (mod π s+1 u L ) for u U L. W can conclud for any λ L, σ(λ) 1 + π s λ L U L. To s this lt λ = u L πl a with p a. Thn σ(λ) λ = σ(u LπL a) u L πl a = σ(u L) u L ( σ(πl ) ) a 1 + π s LU L. ( ) Thus, ν σ(λ) L 1 = s. This implis that G = G λ s and G s+1 = {1}. Thrfor, th uniqu ramification brak occurs at i = s. Now suppos λ is a root of (x) = x p x α, whr α = π m F. Thn, α = λ(λ + 1) (λ + (p 1)) bcaus if λ is a root, thn λ + j for j Z/pZ is a root. In th abov product, (λ + 1),, (λ + (p 1)) ar units, so ν F (α) = ν L (λ). Thrfor, ν F (α) = s. For α = π m F, m = s. Bcaus thr ar infinitly many choics for m, thr ar infinitly many possibl ramification braks, thus xtnsions of dgr p. 8

Not that whn givn two Artin-Schrir polynomials 1 (x) = x p x a and 2 (x) = x p x b for a, b F, ν(a) = ν(b) dos not imply th xtnsions gnratd by 1 and 2 ar isomorphic. If th constant trms a and b diffr by a function of th form c p c for c F, thn 1 and 2 will gnrat isomorphic xtnsions. 5. Exampl W utiliz th rsults provn in th papr to classify all dgr n = 10 fild xtnsions L/F whr F = F p ((T )) with p ±3 (mod 10). W hav L/F is ncssarily on of th following: (1) a dgr 10 unramifid xtnsion, (2) a dgr 2 totally tamly ramifid xtnsion of a dgr 5 unramifid xtnsion, (3) a dgr 5 totally tamly ramifid xtnsion of a dgr 2 unramifid xtnsion, (4) or a dgr 10 totally tamly ramifid xtnsion. From Thorm 3.1, th unramifid portion of ach cas is uniqu. Ths xtnsions ar formd by adjoining a root of th cyclotomic polynomial x pf x and hav Galois group isomorphic to Z/fZ. To comput a dfining polynomial for ths xtnsions, s [5, p. 587] which uss an algorithm to find irrducibl polynomials in th ring F p [x] that can b applid to th polynomial ring ovr F p ((T )). For th totally tamly ramifid portion of th xtnsions, it is ncssary to us a formula similar to th on for th charactristic zro cas outlind in [11]. By Thorm 4.9 thr ar distinct, but not ncssarily non-isomorphic dgr xtnsions. By Thorm 4.10 for g = gcd(, p f 1) thr ar g non-isomorphic totally tamly ramifid xtnsions of dgr and th dfining polynomials ar in th form x ζ r π F for 0 r g 1. Thus for cas 1 thr is 1 uniqu xtnsion and thr ar gcd(2, p 5 1) = 2, gcd(5, p 2 1) = 1, gcd(10, p 1 1) = 2, non-isomorphic xtnsions for cas 2, 3 and 4 rspctivly. In total, thr ar 6 non-isomorphic xtnsions of dgr 10 for such p. To calculat th Galois group of ach of ths xtnsions, it is ncssary to us a lmma found in [14, p. 66-67]: Lmma 5.1. Lt F b a fild of charactristic p. Lt L/F b a Galois xtnsion with Galois group G and lt M L dnot th maximal idal of th intgrs in L. For i 1, lt G i b th i-th ramification group. Lt U 0 b th units in L and for i 1, lt U i = 1 + (πl i ), whr is th gnrator of M L. (a) For i 0, G i /G i+1 is isomorphic to a subgroup of U i /U i+1. (b) Th group G 0 /G 1 is cyclic and isomorphic to a subgroup of th group of roots of unity in th rsidu fild of L. Its ordr is prim to p. (c) Th quotints G i /G i+1 for i 1 ar ablian groups and ar dirct products of cyclic groups of ordr p. Th group G 1 is a p-group. (d) Th group G 0 is th smi-dirct product of a cyclic group of ordr prim to p with a normal subgroup whos ordr is a powr of p. () Th groups G 0 and G ar both solvabl. Th GAP packag [7] in Sag [13] can b usd to find possibl Galois groups as dscribd for xtnsions of Q p in [2 4]. For small dgrs, th onlin L-functions and Modular Forms Databas (LMFDB) [1] can also b usd to find possibl Galois groups with th ncssary proprtis. Th sam tchniqu in finding th Galois group for th p-adic cas can b applid 9

to th function fild cas. Considr on of th cas 2 xtnsions. As mntiond abov, on can us th mthods dscribd in [5, p. 587] to fficintly find a dfining polynomial for /F. For xampl, w find that x 5 + 2x + 1 is a dfining polynomial for /F in th cas p = 3. By Thorm 4.10 dfining polynomials for th two non-isomorphic cas 2 xtnsions ar givn by x 2 T and x 2 ζt whr T is a uniformizr in F and consquntly a uniformizr for /F and ζ is a primitiv p 5 1-st root of unity. W will us Lmma 5.1 to discuss th proprtis of th Galois group and find th Galois group for a cas 2 xtnsion with x 2 T bing a dfining polynomial for L/. Th Galois group of L/ is a solvabl subgroup of S n, or in this cas S 10. Thr ar 24 solvabl subgroups of S 10. Th Galois group will hav a subfild corrsponding to G/G 0, th Galois group of th unramifid intrmdiat xtnsion. This G/G 0 must b isomorphic to Z/5Z sinc th Galois group of an unramifid xtnsion is always isomorphic to Z/fZ. From Lmma 5.1 part (a), G 0 /G 1 is isomorphic to Aut(L/) which is ncssarily isomorphic to Z/2Z sinc L/ is a dgr two xtnsion. Not that Z/2Z is cyclic and of ordr prim to 5. In this particular cas, sinc G i is isomorphic to th trivial group for i 1, G 0 = G0 /G 1. Thus th Galois group must hav a normal subgroup isomorphic to Z/2Z. Th only group which fits ths critria is Z/10Z. Blow is a tabl listing th Galois groups for all six dgr 10 xtnsions: Cas f Gal(L/F ) 1 1 10 Z/10Z 2 2 5 Z/10Z 2 2 5 Z/10Z 3 5 2 F 5 4 10 1 F 5 Z/2Z 4 10 1 F 5 Z/2Z Not that F 5 is th Frobnius group of ordr 20 which is isomorphic to a smidirct product Z/5Z Z/4Z = Z/5Z Aut(Z/5Z). Th sam mthods of finding th Galois group of L/F can b applid to intrmdiat xtnsions. Th following tabl contains information about th intrmdiat unramifid and totally tamly ramifid xtnsions in th cas that p = 3. Cas f Gal(/F ) Polynomial for /F Gal(L/) Polynomial for L/ 1 1 10 Z/10Z x 10 + 2x 2 + 1 2 2 5 Z/5Z x 5 + 2x + 1 Z/2Z x 2 T 2 2 5 Z/5Z x 5 + 2x 4 + 2x + 2 Z/2Z x 2 ζ 242 T 3 5 2 Z/2Z x 2 + x + 2 F 5 x 5 T 4 10 1 F 5 Z/2Z x 10 T 4 10 1 F 5 Z/2Z x 10 ζ 2 T Rfrncs [1] LMFDB. Onlin databas, 2012. [2] C. Awtry. Dodcic 3-adic Filds. Int. J. Num. Th., 8:933 944, 2012. [3] C. Awtry and T. Edwards. Dihdral p-adic filds of prim dgr. Int. J. Pur App. Math., 75:185 194, 2012. [4] Chad Awtry. On Galois Groups of Totally and Tamly Ramifid Sxtic Extnsions of Local Filds. Int. J. Pur App. Math., 70:855 863, 2011. [5] D. Dummitt and R. Foot. Abstract Algbra. John Wily and Sons, Inc., 3rd dition, 2004. 10

[6] I.B. Fsnko and S.V. Vostokov. Local filds and thir xtnsions, volum 121 of Translations of Mathmatical Monographs. Amrican Mathmatical Socity, 2002. [7] Th GAP Group. GAP Groups, Algorithms, and Programming, Vrsion 4.5.5, 2012. [8] F. Gouvêa. p-adic numbrs: An Introduction. Univrsitxt. Springr, Nw York, 2nd dition, 1997. [9] J. Jons and D. Robrts. A databas of local filds. J. Symbolic Comput., 41:80 97, 2006. [10] S. Lang. Algbra, volum 211 of Graduat Txts in Mathmatics. Springr-Vrlag, Brlin, 2004. [11] S. Pauli and X-F. Roblot. On th computation of all xtnsions of a p-adic fild of a givn dgr. Math. Comp, 70(236):1641 1659, 2001. [12] Sbastian Pauli. Efficint Enumration of Extnsions of Local Filds with Boundd Discriminant. PhD thsis, Concordia Univrsity, 2001. [13] SAGE Mathmatics Softwar Vrsion 2.6. http://www.sagmath.org/. [14] J-P. Srr. Local filds, volum 67 of Graduat Txts in Mathmatics. Springr-Vrlag, Nw York, 1979. Dpartmnt of Mathmatical Scincs, Clmson Univrsity, Clmson, SC 29634 E-mail addrss: jimlb@clmson.du Molloy Collg, Rockvill Cntr, NY 11571 E-mail addrss: ahasmani09@lions.molloy.du Univrsity of North Dakota, Grand Forks, ND 58202 E-mail addrss: lindsy.hiltnr@gmail.com Bthany Luthran Collg, Mankato, MN 56001 E-mail addrss: angla.kraft@blc.du Grov City Collg, Grov City, PA 16127 E-mail addrss: scofilddr1@gcc.du Dpartmnt of Mathmatical Scincs, Clmson Univrsity, Clmson, SC 29634 E-mail addrss: kirstiw@g.clmson.du 11