GALOIS STRUCTURE ON INTEGRAL VALUED POLYNOMIALS

Transcription

1 GALOIS STRUCTURE ON INTEGRAL VALUED POLYNOMIALS BAHAR HEIDARYAN, MATTEO LONGO, AND GIULIO PERUGINELLI Abstract. W charactriz finit Galois xtnsions K of th fild of rational numbrs in trms of th rings Int Q (), rcntly introducd by Lor and Wrnr, consisting of thos olynomials which hav cofficints in Q and such that f(). W also addrss th roblm of constructing a basis for Int Q () as a Z-modul. 1. Introduction Th main objct of this ar is to study th class of rings Int Q ( ) := Int( ) Q[X] whr K varis among th st of finit Galois xtnsions of Q; hr is th ring of algbraic intgr of K and Int( ) is th ring of olynomials f K[X] such that f( ). Th rings Int Q ( ) hav bn introducd in [LW1] and studid also in [Pr14]. Among th othr things, th authors of [LW1] rovd that Int Q ( ) is a Prüfr domain. It is immdiat to s that Int Q ( ) is containd in Int(Z) = {f Q[X] f(z) Z}, th classical ring of intgr-valud olynomials. Morovr, if K is a ror fild xtnsion of Q, thn Int Q ( ) is rorly containd in Int(Z): in fact, lt Z b a rim which is not totally slit in ; thn it is not difficult to s that th olynomial f(x) = X(X 1)... (X ( 1)) is in Int(Z) \ Int Q ( ). This is an vidnc of th fact that, for th class of finit Galois xtnsion K/Q, th ring Int Q ( ) is comltly dtrmind by th st of rim Z which ar totally slit in, and thrfor by th fild K itslf. Our main rsult is a charactrization of finit Galois xtnsion of Q in trms of ths rings Int Q ( ). Mor rcisly, as a corollary of our main rsult Thorm.7, w rov th following: Thorm 1.1. Lt K and K b finit Galois xtnsions of Q. Thn Int Q ( ) = Int Q ( ) if and only if K = K. Th statmnt is fals if w considr finit xtnsions of Q which ar not Galois. In fact, if K/Q is a finit non-galois xtnsion and K is any conjugat fild of K ovr Q diffrnt from K, thn it is asy to s that Int Q ( ) = Int Q ( ). W can rformulat th main rsult in mor abstract trms as follows. Dnot G th catgory whos objcts ar ring of intgrs of finit Galois xtnsions K/Q with homomorhism givn by inclusions, and by C th catgory of subrings of Q[X] in which morhisms ar again inclusions. Thn th functor Int Q : G C Dat: Novmbr, Mathmatics Subjct Classification. 13F0, 11R3, 11S0, 11C08. Ky words and hrass. Charactristic idal; Finit Galois xtnsion; Intgr-valud olynomial; Rgular basis. 1

2 B. HEIDARYAN, M. LONGO, AND G. PERUGINELLI which taks an objct of G to Int Q ( ) and th inclusion to th inclusion Int Q ( ) Int Q ( ), is a faithful contravariant functor. W nxt addrss th roblm of constructing a rgular basis of Int Q ( ) as a Z-modul. In articular, w discuss th valu of th -adic valuation of th lading trm of th lmnt of dgr n in rgular basis, for ach rim numbr. W show that this is quivalnt to undrstand th analogu local qustion of dtrmin th -adic valuation of Int Q (K) := Int( ) Q [X] for ach finit xtnsion K/Q, whr Int( ) is th ring of f K[X] such that f( ), and is th valuation ring of K. W comltly dtrmin ths valus in Thorm 3., in th cas of tam ramification. As a consqunc, w obtain th scond main rsult of this ar. To stat th thorm, lt K/Q b a Galois xtnsion and, for any rim of Z, lt q and b th cardinality of th rsidu fild of any rim idal of abov and th ramification indx of, rsctivly. W also st w q (n) = n j 1 q j and dfin for vry intgr n 1, wq (n) ω (n) = ω K, (n) :=. Thorm 1.. Suos that K/Q is a Galois xtnsion which is tamly ramifid at ach rim. Lt {f n (X)} n 0 b a Z-basis of Int Q ( ) such that dg(f n ) = n, for ach n N. Thn w can writ f n (X) = g n(x) ω(n) for som monic olynomial g n (X) in Z[X], whr th roduct is ovr all rims of Z. Th roof of th abov thorm is constructiv: first, w construct a basis of Int Q (v ), for any rim v of K, from th knowldg of local basis of Int(v ); thn, w us th Chins Rmaindr Thorm to construct a global basis of Int Q ( ).. A charactrization of Galois xtnsion W introduc th following gnral notation, xtnding that of th introduction. Lt D b an intgral domain with quotint fild K and lt A b a torsion-fr D-algbra. Lt B := A D K; w hav a canonical mbdding A B and K B. For a A and f K[X], th valu f(a) blongs to B, and th following dfinition maks sns (s also [PW14]): Int K (A) := {f K[X] : f(a) A, a A}. Clarly, Int K (A) is a D-algbra. It is asy to s that Int K (A) is containd in th classical ring of intgr-valud olynomials Int(D) = {f K[X] f(d) D} if and only if A K = D, and this will b th cas hncforth. A squnc of olynomials {f n (X)} n N Int K (A) which form a basis of Int K (A) as a D-modul and such that dg(f n ) = n for ach n N, is calld rgular basis of Int K (A). W dfin I n (Int K (A)) to b th D-modul gnratd by th lading cofficints of all th olynomials f Int K (A) of dgr xactly n; w call ths D-moduls charactristic idals. For ach n N, by th abov assumtion and [CC97, Proosition II.1.1], I n (Int K (A)) is a fractional idal of D. Morovr, th st of charactristic idals forms an ascnding squnc: D I 0 (Int K (A))... I n (Int K (A)) I n+1 (Int K (A))... K. Th link btwn rgular bass and charactristic idals is givn by [CC97, Proosition II.1.4], which says that a squnc of olynomials {f n (X)} n N of Int K (A) is a rgular basis if and

3 only if, for ach n N, f n (X) is a olynomial of dgr n whos lading cofficint gnrats I n (Int K (A)) as a D-modul. In articular, not that Int Q ( ) and Int Q ( ) (for K/Q and K/Q finit fild xtnsions) admit rgular basis. W fix from now on to th nd of this Sction a numbr fild K and dnot by its ring of algbraic intgrs. For any rim idal of, w dnot,() th localization of at, i.., th localization at th multilicativ st \. Morovr, for any Z-modul M and any rim numbr, w dnot M () th localization at, i.., th localization at th multilicativ st Z \ Z. W also dnot K th comltion of K at and, th valuation ring of. Proosition.1. Int Q ( ) = Int Q(,() ) and 3 I n (Int Q ( )) = I n (Int Q (,() ) whr th intrsction is ovr all rim idals of K. Proof. W first obsrv that Int Q ( ) = Int Q( ) () ; hr th intrsction is ovr all rims of Z. Thn on obsrvs that Int Q ( ) () = Int Q (,() ) (s for xaml [Wr14]). W conclud that I n (Int Q (, ) () ) is qual to I n (Int Q (,() ), showing th scond art. Furthr,,() =,(), whr,() is th localization of at, and th intrsction is ovr all rim idals of K which li abov. Thrfor (1) Int Q (,() ) = and th rsult follows. Int Q (,() ) Rmark.. Not that, if K/Q is Galois, thn Int Q (,() ), for, ar all qual bcaus Gal(K/Q) acts transitivly on th st of rings {,() : }. Thrfor (1) rads as for ach. Int Q (,() ) = Int Q (,() ) In ordr to dtrmin som rlation of containmnts btwn th rings Int Q (,() ), w introduc th following notation: givn an xtnsion of commutativ rings R S, w considr th null idal of S ovr R: N R (S) = {g R[X] g(s) = 0} R[X]. Proosition.3. Lt K b a numbr fild and lt Z b a rim. Lt b a rim idal abov with ramification indx and rsidu class dgr f. Thn N F ( / ) = ((X f X) ) Proof. Sinc π : / /P 1... / = F f and F mbds in all of ths rings (bcaus i Z = Z = Z, for all i = 1,..., ) w hav / / 1... / F so, in articular, w hav th following chain of containmnts btwn ths idals of F [X]: N F ( / ) N F ( / 1 )... N F ( /). Sinc / is a finit fild with f lmnts, th idal N F ( /) is gnratd by X f X. Th roof rocds by induction on. Suos that N F ( / 1 ) is gnratd by (X f X) 1. It is asy to s that (X f X) is containd in N F ( / ). Thrfor, th lattr

4 4 B. HEIDARYAN, M. LONGO, AND G. PERUGINELLI idal is gnratd by a olynomial g F [X] which is zro on all th lmnts of / of th form g(x) = (X f X) 1 h(x) = F q (X) 1 γ S(X γ) for som S F f = F q. Suos that S is strictly containd in F q and lt γ F q \ S. Without loss of gnrality, w may assum that γ = 0 (aly th automorhism X X γ, if ncssary; this is an automorhism for F f and / ). Lt t P/P / such that its indx of nilotncy is (that is, t = 0 but t 1 0). Thn F q (t) 1 = t 1 (t q 1 1) 1 is not zro in /, bcaus t q 1 1 is a unit of / (bcaus / is th Jacobson radical of / ). In th sam way, h(t) = γ S (t γ) is not in th krnl of π : / / 1, which is /, bcaus modulo, h(t) is not zro (π(h(t)) = h(π(t)) = h(0) 0, bcaus 0 / S). Hnc, h(t) is invrtibl, so that g(t) = F q (t) 1 h(t) is not zro, contradiction. Proosition.4. Lt K, K b numbr filds, with rim idals,, rsctivly, with ramification indx/rsidu class dgr qual to, f and, f, rsctivly. Suos that Int Q (,( )) Int Q (,() ) Thn Z = Z = Z, f f and. In articular, if th abov containmnt is an quality, w hav that Z = Z = Z, f = f and =. Proof. Suos that Z = Z and Z = Z. Obsrv that Int Q (,() ) Q = (Int(,() ) K) Q =,() Q = Z () and analogously for and. Thrfor Int Q (,( )) Q = Z ( ) Int Q (,() ) Q = Z (). Hnc, =. By Proosition.3, th containmnt of th hyothsis imlis that () In articular, modulo, w hav (X f X) Int Q (,() ). (X f X) N F ( / ) = ((X f X) ), again by Proosition.3. It follows that (X f X) (X f X) and sinc th lattr is a radical idal (bcaus X f X is a sarabl olynomial), this mans that X f X (X f X) which is quivalnt to F f F f which holds if and only if f f, as claimd. In th sam way, sinc X f X is a sarabl olynomial (vry irrducibl factor aars with multilicity 1 in th factorization of X f X ovr F ), w dduc that. W rcall that, by a rsult of Grboud (s [Gr93] and also [CC97, Pro. IV.3.3]) w hav (3) Int(Z (),,() ) = {f K[X] f(z () ),() } = Int(Z () ),() Lmma.5. Lt K b a numbr fild and lt b a rim idal which lis abov a rim Z. Lt = ( ) and f = f( ) b th ramification indx and rsidu class dgr, rsctivly. Thn th following conditions ar quivalnt: i) Int(Z () ) Int(,() ). ii) Int(Z (),,() ) = Int(,() ) iii) Int Q (,() ) = Int(Z () ).

5 iv) = f = 1. If any of this quivalnt conditions holds, thn Int(Z () ),() = Int(,() ). Proof. Obviously, conditions i) and iii) ar quivalnt, sinc w always hav Int Q (, ) Int(Z () ). If i) holds, thn by (3) abov w hav Int(Z (),, ) Int(, ), which is th condition ii), sinc w always hav th containmnt Int(Z (),, ) Int(, ). Convrsly, if condition ii) holds, thn again by (3) abov w hav Int(Z () ) Int(, ). Th quivalnc btwn iii) and iv) follows immdiatly from Proosition.4. Corollary.6. Lt K b a numbr fild and lt Z b a rim. conditions ar quivalnt: i) Int(Z () ) = Int Q (,() ). ii) is totally slit in. iii) X X Int Q ( ). 5 Thn th following Proof. Th roof of th quivalnc i) ii) follows immdiatly from (1) and Lmma.5. Indd, if is totally slit in thn, for ach rim idal of abov, w hav Int(Z () ) = Int Q (,() ), so that by (1) w hav th quality Int(Z () ) = Int Q (,() ). Convrsly, if th last quality holds, thn by (1), for ach rim idal of abov, w hav Int(Z () ) Int Q (,() ) Int(Z () ), so quality holds throughout and is totally slit in. W show now that ii) iii). Suos that is totally slit in, so that, by th Chins Rmaindr Thorm w hav / = F n whr n = [K : Q]. Hnc, X X is zro on /, so that f(x) = X X is in Int Q ( ). Convrsly, suos that f(x) is in Int Q ( ). Thn X X is zro on / = g i=1 / i i, whr 1,..., g ar th rim idals of abov, with ramification indx i = ( i ) and rsidu class dgr f i = f( i ). Consquntly, X X is zro on ach factor ring / i i, for i = 1,..., g. Lt α b in th Jacobson idal of / i i, that is, α is in i/ i i (th uniqu maximal idal of / i i ). Thn 1 α 1 is a unit in / i. But by assumtion α α = α(α 1 1) = 0, so that α = 0. Thrfor, / i i has trivial Jacobson idal, which hans rcisly whn i = 1. If f i > 1, thn / i is a ror finit fild xtnsion of F, so if w tak an lmnt γ of / i \ F, γ will b a zro of a monic irrducibl olynomial q(x) ovr F of dgr strictly largr than 1. Sinc X X is zro on γ, w would hav that q(x) divid X X ovr F, which is clarly not ossibl bcaus X X slits ovr F. This shows that iii) ii). Th nxt rsult charactrizs th finit Galois xtnsions of Q in trms of th rings Int Q ( ). In articular, w can rcovr from Int Q ( ), if K/Q is Galois. Givn a subring R of Q[X], for ach α Z w considr th following subst of Q(α): R(α) = {f(α) f R} Thorm.7. Lt K/Q b a finit xtnsion and lt R = Int Q ( ). Thn K/Q is a Galois xtnsion {α Z R(α) Z} =. In articular, if K and K ar two Galois xtnsions of Q such that Int Q ( ) = Int Q ( ), thn K = K. Not that th condition R(α) Z is quivalnt to R(α) O Q(α). i

6 6 B. HEIDARYAN, M. LONGO, AND G. PERUGINELLI Proof. Th scond statmnt about K and K follows immdiatly from th first. For th first statmnt, lt R = Int Q ( ) and suos that {α Z R(α) Z} =. It is asily sn that th lft-hand sid is invariant undr th action of th absolut Galois grou Gal(Q/Q). Hnc, contains th ring of intgrs of all th conjugats of K ovr Q, so K/Q is Galois. Convrsly, suos that K/Q is a Galois xtnsion. It is clar that w hav th containmnt {α Z R(α) Z}. Convrsly, lt α Z, α /. W hav to show that thr xists f Int Q ( ) such that f(α) / Z. Lt K α = Q(α) and lt N α b th Galois closur of K α ovr Q (th comositum insid Q of all th conjugats ovr Q of K α ). W hav that α / K K α K N α K, whr th last quivalnc holds bcaus by assumtion K/Q is Galois. By Tchbotarv s Dnsity Thorm, a Galois xtnsion K of Q is comltly dtrmind by th st of rims S(K/Q) which ar totally slit in K (s [Nu99, Chatr VII, Corollary 13.10]). Hnc, th condition N α K is quivalnt to S(K/Q) S(N α /Q), that is, th st of rims Z which ar totally slit in K is not containd in th st of rims which ar totally slit in N α. Lt Z b such a rim and suos also that - is not ramifid nithr in K nor in N α. - dos not divid [α : Z[α]] Th abov rims ar always finit in numbr and sinc th abov st is infinit, by rmoving th lattr rims w still gt a non-mty st. By Corollary.6, f(x) = X X is in Int Q ( ) but not in Int Q (O Nα ). Rcall that a rim Z slits comltly in th normal closur N α of K α (ovr Q) if and only if it slits comltly in K α ([Mar77, Chat. 4, Corollary of Thorm 31]). Hnc, thr xists som rim idal of α abov which has inrtia dgr strictly gratr than 1. Sinc dos not divid [α : Z[α]], it follows by Ddkind- Kummr s Thorm (s [Nu99, Chatr I, Proosition 8.3]) that th factorization in F [X] of th rsidu modulo of th minimal olynomial α (X) of α ovr Z has at last on irrducibl olynomial ovr F whos dgr is strictly gratr than 1; this factor corrsonds to a rim idal of α abov which is not inrt, that is α / F. In articular, this mans that modulo, α is not in F, and so it is not annihilatd by g(x) = X X (quivalntly, modulo, α is a zro of an irrducibl olynomial ovr F of dgr strictly gratr than 1). This imlis that f(α) is not intgral ovr Z. Rmark.8. W also offr a shortr roof of th scond statmnt in Thorm.7: If K and K ar two Galois xtnsions of Q such that Int Q ( ) = Int Q ( ), thn K = K. Suos th abov assumtion is satisfid. In articular, for ach rim Z, if w localiz at Z \ Z w hav th following quality: (4) Int Q (,() ) = Int Q (,()) Lt now Z b a rim which is totally slit in. Thn th lft hand sid of (4) is qual to Int(Z () ), by Corollary.6. By th sam Corollary, is totally slit in. Symmtrically, if is totally slit in K w dduc in th sam way that is totally slit in K. Thrfor, th sts of rims Z which ar totally slit in th Galois xtnsions K and K, rsctivly, coincid. By th Tchbotarv Dnsity Thorm (s [Nu99, Chat. VII, 13, Corollary 3.10]), a finit Galois xtnsion K is uniquly dtrmind by th st of rims Z which ar totally slit in, so K = K. 3. Charactristic idals Proosition.1 rducs th study of charactristic idals of Int Q ( ) to th study of charactristic idals in th local cas. W will addrss a dscrition of ths idals and aly th local rsults to th global contxt.

7 3.1. Local cas. Fix a finit fild xtnsion K/Q having rsidu class dgr f and ramification dgr. Dnot v th -adic valuation of Q, normalizd such that v () = 1. Lt: w (n) := v (n!) = n j j 1 and, if q = f is th cardinality of th rsidu fild of K, ut w q (n) := n q j. j 1 Th following quality follows from [CC97, Corollary II..9]: and, similarly, w hav: v (I n (Int(Z ))) = w (n) (5) v π (I n (Int( ))) = w q (n) whr π is a uniformizr of K and v π th associatd valuation. W dfin finally w Q (n) := v ( In ( IntQ ( ) )). Th following quality holds bcaus of th nxt Lmma, noticing that n = n : I n (Int( )) Q = wq(n) Z and sinc I n (Int Q ( )) I n (Int( )) Q, for vry n N w hav: (6) w Q wq (n) (n) Lmma 3.1. Lt n Z and = ( ), whr is th maximal idal of. Thn n Q = n Z Proof. ( ). Clarly, n n v ( n ) = n n, which is tru, so th containmnt follows, sinc clarly n Q is a Z -modul. ( ). Lt α n Q, say α = m u, whr u Z and m = v (α). Thn v (α) = m which has to b gratr than or qual to n. Thrfor, m n, so α n Z. Th main rsult of this sction shows th oosit inquality in (6) in th cas of tam ramification for a finit Galois xtnsion. By th abov rmarks, this corrsonds to say that I n (Int Q ( )) = I n (Int( )) Q, for ach n N. W show in Examls 3.6 that ths two conditions, namly, Galois and tam ramification, cannot b rlaxd. Thorm 3.. Lt K/Q b a finit tamly ramifid Galois xtnsion, with ramification indx and rsidu fild of cardinality q. Thn for all n N w hav w Q wq (n) (n) =. In articular, w Q (n) only dnds on n, q and. wq(n) Proof. By (6) it is sufficint to show that d = is in I n (Int Q ( )). W obsrv that if f(x) = n i=0 a ix i blongs to Int( ), thn f σ (X) := n i=0 σ(a i)x i blongs to Int( ) for all σ G = Gal(K/Q ) (hr w us crucially th assumtion that K/Q 7

8 8 B. HEIDARYAN, M. LONGO, AND G. PERUGINELLI is Galois). As a consqunc, if w dnot tr = tr K/Q : K Q th trac homomorhism, w s that Tr(f) := n f σ = tr(a i )X i σ G blongs to Int Q ( ), if f Int( ). Thrfor, th trac homomorhisms btwn th function filds Tr : K(X) Q (X) rstricts to Tr : Int( ) Int Q ( ). Sinc, th trac homomorhism tr is surjctiv (th convrs is also tru, s [Nar04, Chatr 5, Corollary,. 7]). Fix α such that tr(α) = 1. Lt c = dα K. In articular, sinc th trac is a Q -homomorhism, w hav tr(c) = d. Not that th v π -valu of c is gratr than or qual to wq(n) w q (n). By (5), c is in I n (Int( )), so thr xists f Int( ) of dgr n whos lading cofficint is qual to c. Thrfor, Tr(f) is a olynomial of dgr n in Int Q ( ) with lading cofficint qual to d, as w wantd to show. Rmark 3.3. W rmark that, from th fact that tr = tr K/Q : Z is surjctiv (bcaus th xtnsion is tam), th roof of Thorm 3. also shows that th rstriction of th trac homomorhism Tr : Int( ) Int Q ( ) is surjctiv. In fact, for ach n N, th n-th lmnt of a rgular basis of Int Q ( ), whos lading cofficint has -adic valu i=0 wq(n) by th abov Thorm, is th imag via th trac homomorhism of a olynomial of Int( ). Obviously, if Tr is surjctiv, it is asily sn that tr is surjctiv, bcaus Z Int Q ( ). Finally, w hav th following commutativ diagram: Int( ) Tr Int Q ( ) tr Z Th nxt corollary shows that Thorm.7 is fals in th local cas. Corollary 3.4. Lt K 1, K b two finit tamly ramifid Galois xtnsions of Q. Thn Int Q (1 ) = Int Q ( ) if and only if K 1 and K hav th sam ramification indx and rsidu fild dgr. Proof. Suos that K 1 and K hav th sam ramification indx and rsidu fild dgr. In articular, th functions w Q i (n), for i = 1,, ar th sam, by Thorm 3.. Hnc, by dfinition, th st of charactristic idals of th rings Int Q (i ), i = 1,, coincid, so ths rings hav a common rgular bass, and thrfor thy ar qual. Convrsly, if th Int Q -rings ar qual, a straightforward adatation of Proosition.4 to th rsnt stting shows that th ramification indxs and rsidu fild dgrs of K 1 and K ar th sam. Not that this art of th roof holds also without th tamnss assumtion. Rmark 3.5. In th cas K/Q is a finit unramifid xtnsion (so, in articular, a Galois xtnsion), w can givn an xlicit basis of Int Q ( ). Lt q = f b th cardinality of th rsidu fild of. By Thorm 3., for all n N w hav w Q (n) = w q (n). Lt f(x) := Xq X which clarly blongs to Int Q ( ). For k N, w dnot by f k (X) th comosition of f with itslf k tims, namly f k (X) = f... f(x). If k = 0 w ut f 0 (X) := X. For ach ositiv intgr n N, w considr its q-adic xansion: n = n 0 + n 1 q n r q r

9 whr n i {0,..., q 1} for all i = 0,..., r. W dfin r f n (X) := (f i (X)) n i i=0 Notic that f n (X) = X n for n = 0,..., q 1 and f q (X) = f(x). Morovr, f n Int Q ( ) and has dgr n, for vry n N. It is asy to rov by induction that lc(f i ) = a i, whr a i = 1 + q q i 1 = w q (q i!). By th sam roof of [CC97, Cha., Pro. II..1] on can show that lc(f n ) = wq(n) for vry n N, so, finally, th family of olynomials {f n (X)} n N is a rgular basis of Int Q ( ). Examls 3.6. In th nxt two xamls w show th assumtions in Thorm 3. cannot b drod. (1) If K/Q is not a Galois xtnsion, thn th rstriction of th trac homomorhism to Int( ) may giv a olynomial in Q (X) which is not in Int Q ( ). For xaml, lt K = Q ( 3 ), whos ring of intgrs is = Z [ 3 ]. Thn th olynomial f(x) = X(X 1)(X 3 )(X (1 + 3 )) is in Int( ) but its trac ovr Q (X) is qual to g(x) = 3X (X 1), which is not intgrvalud ovr, sinc g( 3 ) /. On can show by an xlicit comutation that in this wq(n) xaml th quality w Q (n) = lmnts of a -basis of Int( ) ar w (4) 3 f 1 (X) = X; f (X) = dos not hold for n = 4. X(X 1) 3 ; f 3 (X) = X(X 1)(X 3 ) 3 ; 9 Indd, th first four f 4 (X) = X(X 1)(X 3 )(X (1 + 3 )) ; and considring all ossibl -combinations of ths lmnts which li in Q [X], w s that thr is no lmnt in Int Q ( ) of dgr 4 whos lading cofficint has valuation 1 =. () W now discuss th tamnss assumtion. Considr th cas of K = Q (i) with i = 1 and lt {f n (X) : n 0} b a rgular basis of Int( ) obtaind by mans of comositions and roduct of th Frmat olynomial X X 1+i (in th sam way as in th Examl 3.5; s [CC97, Chatr II,. 3]). W st G(X) = X X. On can chck that and f 6 + if 4 = G3 4 + G G f 10 + f 8 if 6 + (1 i)f 4 = G G3 8 G 4 + G blong to Int Q ( ) and thir lading cofficints hav valuation qual to and w (10) = 4, rsctivly; on can also chck that w (n) v (I n (Int Q ( ))) = w (6) = for all n 11. On th othr hand, writing down a basis of Int( ) u to dgr 1, and considring all ossibl -combinations of ths lmnts which li in Q [X], w s that w (1) v (I 1 (Int Q ( ))) = 1.

10 10 B. HEIDARYAN, M. LONGO, AND G. PERUGINELLI It might b intrsting to dscrib th valus takn by v ( In ( IntQ ( ) )) in th cas of wild ramification. 3.. Global cas. Lt K/Q b a finit Galois xtnsion with absolut discriminant D and dgr d ovr Q. For ach rational rim, dnot f th rsidu class dgr and its ramification dgr. Lt q = f b th cardinality of th rsidu fild of K. Th following is a rformulation of Thorm 1. in th Introduction: Thorm 3.7. If K/Q is Galois such that for all rational rims,. Thn ( I n (Int Q ( )) = wq ) (n) as fractional idals of Z, whr th roduct is ovr all rational rims and, for ach rim, w choos a rim idal. Proof. Not that for a fixd n w hav w q (n) for almost all rim owrs q, and thrfor th abov roduct is wll dfind. Th rsult follows immdiatly combining Proosition.1 and Thorm 3. Rfrncs [CC97] P.J. Cahn and J.L. Chabrt, Intgr-valud olynomials, Mathmatical Survys and Monograhs, vol. 48, Amrican Mathmatical Socity, Providnc, RI, MR (98a:1300) [Gr93] G. Grboud, Substituabilité d un annau d Ddkind, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 1, 9 3. MR (94:13039) [LW1] K.A. Lor and N.J. Wrnr, Gnralizd rings of intgr-valud olynomials, J. Numbr Thory 13 (01), no. 11, MR [Mar77] D.A. Marcus, Numbr filds, Sringr-Vrlag, Nw York-Hidlbrg, 1977, Univrsitxt. MR (56 #15601) [Nar04] W ladys law Narkiwicz, Elmntary and analytic thory of algbraic numbrs, third d., Sringr Monograhs in Mathmatics, Sringr-Vrlag, Brlin, 004. MR (005c:11131) [Nu99] J. Nukirch, Algbraic numbr thory, Grundlhrn dr Mathmatischn Wissnschaftn [Fundamntal Princils of Mathmatical Scincs], vol. 3, Sringr-Vrlag, Brlin, 1999, Translatd from th 199 Grman original and with a not by Norbrt Schaachr, With a forword by G. Hardr. MR (000m:11104) [Pr14] Giulio Pruginlli, Intgral-valud olynomials ovr sts of algbraic intgrs of boundd dgr, J. Numbr Thory 137 (014), MR [PW14] Giulio Pruginlli and Nicholas J. Wrnr, Intgral closur of rings of intgr-valud olynomials on algbras, Commutativ algbra, Sringr, Nw York, 014, MR [Wr14] N.J. Wrnr, Int-dcomosabl algbras, J. Pur Al. Algbra 18 (014), no. 10, MR B. H. Diartimnto di Matmatica, Univrsità di Padova, Via Trist 63, 3511 Padova, Italy, and Dartmnt of Mathmatics, Tarbiat Modars Univrsity, , Thran, Iran. addrss: b.hidaryan@modars.ac.ir M.L. Diartimnto di Matmatica, Univrsità di Padova, Via Trist 63, 3511 Padova, Italy addrss: mlongo@math.unid.it G. P. Diartimnto di Matmatica, Univrsità di Padova, Via Trist 63, 3511 Padova, Italy addrss: grugin@math.unid.it