ON DEDEKIND’S CRITERION AND MONOGENICITY OVER DEDEKIND RINGS

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PII. S0161171203211534 http://ijmms.hindawi.com

ON DEDEKIND’S CRITERION AND MONOGENICITY OVER DEDEKIND RINGS

M. E. CHARKANI and O. LAHLOU Received 29 November 2002

We give a practical criterion characterizing the monogenicity of the integral closure of a Dedekind ringR, based on results on the resultant Res(P , Pi)of the minimal polynomialPof a primitive integral element and of its irreducible factorsPi modulo prime ideals ofR. We obtain a generalization and an improvement of the Dedekind criterion (Cohen, 1996) and we give some applications in the case where Ris a discrete valuation ring or the ring of integers of a number ﬁeld, generalizing some well-known classical results.

Mathematics Subject Classiﬁcation: 11Y40, 13A18, 13F30.

1. Introduction. LetKbe an algebraic number field and letOKbe its ring of integers. IfOK=Z[θ]for some numberθinOK, we say thatOK has a power basis orOKis monogenic. The question of the existence of a power basis was originally examined by Dedekind [5]. Several number theorists were interested in and attracted by this problem (see [7, 8, 9]) and noticed the advantages of working with monogenic number fields. Indeed, for a monogenic number fieldK, in addition to the ease of discriminant computations, the factorization of a primepinK/Qcan be found most easily (see [4, Theorem 4.8.13, page 199]). The main result of this paper isTheorem 2.5which characterizes the monogenicity of the integral closure of a Dedekind ring. More precisely, letR be a Dedekind domain,Kits quotient field,La finite separable extension of degreenofK,αa primitive element ofLintegral overK,P (X)=Irrd(α, K), ma maximal ideal ofR, andOLthe integral closure ofRinL. Assume that P (X)¯ =r

i=1P¯_i^eⁱ(X)in(R/m)[X]withei≥2, and letPi(X)∈R[X]be a monic lifting of ¯Pi(X)for 1≤i≤r. Then we prove thatOL=R[α]if and only if, for every maximal idealmofRandi∈ {1, . . . , r},vm(Res(Pi, P ))=deg(Pi), where vmis them-adic discrete valuation associated tom. This leads to a necessary and suﬃcient condition for a simple extensionR[α]of a Dedekind ringRto be Dedekind. At the end, we give two illustrations of this criterion. In the second example, we give the converse which was not known yet.

2. Monogenicity over a Dedekind ring. Throughout this paperRis an integral domain,Kits quotient ﬁeld,Lis a ﬁnite separable extension of degree nofK,αis a primitive element ofLintegral overR,P (X)=Irrd(α, K),mis

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a maximal ideal ofR, andOLis the integral closure ofRinL. Letf andgbe two polynomials overR; the resultant off andgwill be denoted by Res(f , g) (see [11]).

Definition2.1. IfOL=R[θ]for some numberθ∈OL, thenOLhas a power basis orOLis monogenic.

Proposition2.2. LetRbe an integrally closed ring and letαbe an integral element overR. Then(R[α])p=Rp[α]for every prime idealpofR. In partic- ular,OL=R[α]if and only ifRp[α]is integrally closed for every prime idealp ofRif and only ifR[α]is integrally closed.

Proof. We obtain the result from the isomorphismR[α]R[X]/P (X), the properties of an integrally closed ring and its integral closure, and the properties of a multiplicative closed subset of a ringR, notably,S⁻¹(R[X])= (S⁻¹R)[X](see [1]).

Definition2.3. LetRbe a discrete valuation ring (DVR),p=π Rits maximal ideal, andαan integral element overR. LetPbe the minimal polynomial ofα, and ¯P (X)=r

i=1P¯_i^eⁱ(X)the decomposition of ¯P into irreducible factors in(R/p)[X]. Set

f (X)= r i=1

Pi(X)∈R[X],

h(X)= r i=1

P_i^eⁱ⁻¹(X)∈R[X],

T (X)=P (X)−r

i=1P_i^eⁱ(X)

π ∈R[X],

(2.1)

wherePi(X)∈R[X]is a monic lifting of ¯Pi(X), for 1≤i≤r. We will say that R[α]isp-maximal if(f ,¯T ,¯h)¯ =1 in(R/p)[X](where(·,·)denotes the greatest common divisor (gcd)). IfRis a Dedekind ring andpis a prime ideal ofR, then we say thatR[α]isp-maximal ifRp[α]ispRp-maximal.

Remarks2.4. (1) Ifπ is uniramiﬁed inR[α], that is,ei=1 for all i, then h¯=¯1 and thereforeR[α]isp-maximal.

(2) Let π be ramiﬁed in R[α], that is, there is at least one i such that ei≥2. LetS = {i∈ {1, . . . , r} |ei≥2}andf1(X)=

i∈SPi(X)∈R[X]. Then (f¯1,T )¯ =(T ,¯f ,¯h)¯ in(R/p)[X]since ¯f1=(f ,¯h). In particular, if every¯ ei≥2, then(f ,¯T )¯ =(T ,¯ f ,¯h), because ¯¯ fdivides ¯hin this case.

(3)Deﬁnition 2.3is independent of the choice of the monic lifting of the ¯Pi. More precisely, let

P (X)¯ = r i=1

P¯_i^eⁱ(X)= r i=1

Q¯^e_iⁱ(X) with ¯Pi(X)=Q¯i(X)for 1≤i≤r in(R/p)[X].

(2.2)

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Set

g(X)= r i=1

Qi(X)∈R[X], k(X)= r i=1

Q^e_iⁱ⁻¹(X)∈R[X]

U (X)=π⁻¹



P (X)− r i=1

Q^e_iⁱ(X)



∈R[X].

(2.3)

Then(f ,¯T ,¯h)¯ =1 in(R/p)[X]if and only if(g,¯U ,¯ ¯k)=1 in(R/p)[X]. Indeed, we may assume thatR is a DVR and p=π R. LetV1=(g−f )/π andV2= (k−h)/π. Thenπ T=π U+gk−f h. Replacinggbyπ V1+fandkbyπ V2+h, we ﬁnd that ¯T =U¯+V¯1h¯+V¯2f¯and therefore(T ,¯f ,¯h)¯ =(U ,¯ f ,¯h)¯ =(U ,¯ g,¯k)¯ since ¯f=g¯and ¯h=k.¯

Theorem2.5. LetRbe a Dedekind ring. LetPbe the minimal polynomial of α, and assume that for every prime idealpof R, the decomposition of P¯into irreducible factors in(R/p)[X]verifies:

P (X)¯ = r i=1

P¯_i^eⁱ(X)∈(R/p)[X] (2.4)

withei≥2fori=1, . . . , randPi(X)∈R[X]be a monic lifting of the irreducible factorP¯i for i=1, . . . , r. Then OL=R[α]if only ifvp(Res(Pi, P ))=deg(Pi) for every prime idealpofRand for everyi=1, . . . , r, wherevp is thep-adic discrete valuation associated top.

For the proof we need the following two lemmas.

Lemma2.6. Letp=uR+vRbe a maximal ideal of a commutative ringR.

ThenpRp=vRpif and only if there exista, b∈Rsuch thatu=au²+bv.

Proof. IfpRp=vRp, then there exists∈Randt∈R−psuch thattu=vs.

Sincep is maximal inR, so there existst∈R such thattt−1∈p. Hence u−utt=u−vst∈p² and there exista, b∈R such that u=au²+bv.

Conversely, u²R+vR⊆vR+p²⊆p. If there exist a, b∈R such that u= au²+bv, thenp=u²R+vRand thereforevR+p²=p. Localizing atp and applying Nakayama’s lemma, we ﬁnd thatpRp=vRp.

Lemma2.7. LetR be a commutative integral domain, letKbe its quotient field, and consider P , g, h, T ∈ R[X]. If g is monic and P =gh+π T, then Res(g, P )=π^deg(g)Res(g, T ). In particular, ifm=π R is a maximal ideal of Rand ifP (X)¯ =r

i=1P¯_i^eⁱ(X)is the decomposition ofP¯into irreducible factors in(R/m)[X], withPi(X)∈R[X] a monic lifting ofP¯i(X)for 1≤i≤r, and T (X)=π⁻¹(P (X)−r

i=1P_i^eⁱ(X))∈R[X], then Res

Pi, P

=π^deg(Pⁱ⁾Res Pi, T

(2.5)

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and(P¯i,T )¯ =1in(R/m)[X]if and only if

Res Pi, T

=Res Pi, P

π^deg(Pⁱ⁾ ∈R−m. (2.6)

Proof. Letx1, . . . , xmbe the roots ofgin the algebraic closure ¯KofK. It is then easy to see (see [11]) that Res(g, P )=m

i=1P (xi)=π^deg(g)Res(g, T )be- causeP (xi)=π T (xi). The second result follows from Res(P¯i,P )¯ =Res(Pi, P ) and [2, Corollary 2, page 73].

Proof ofTheorem2.5. By Proposition 2.2, we may assume that R is a DVR. Letp be a prime ideal ofRand(OL)(p) the integral closure ofRp inL.

Let ¯P (X)=Πi^r=1P¯_i^eⁱ(X)in(Rp/pRp)[X]withei≥2 andPi(X)∈Rp[X]a monic lifting of ¯Pi(X)for 1≤i≤r. Let

T (X)=P (X)−Π^ri=1P_i^eⁱ(X)

π ∈Rp[X] (2.7)

withπ Rp=pRp.

(a) We prove that if(P¯i,T )¯ =1 in(Rp/pRp)[X]for everyi=1, . . . , r, then (OL)(p)=Rp[α]=A. Indeed, ¯P (X)=Π^ri=1P¯_iêⁱ(X)in(Rp/pRp)[X]andRpis a local ring, so by [14, Lemma 4, page 29] (see also [3]) the idealsᏮi=π A+Pi(α)A (i=1, . . . , r) are the only maximal ideals ofA, soAis integrally closed if and only ifᏭᏮi is integrally closed for everyi=1, . . . , r. More generally, we prove that every ᏭᏮi is a DVR. SinceRp is Noetherian, so Rp[α]Rp[X]/P (X) is Noetherian, henceᏭᏮi is Noetherian sinceᏭᏮi is a local integral domain with maximal ideal ᏮiᏭᏮi. It remains to show that ᏮiᏭᏮi is principal. In- deed,(P¯i,T )¯ =1 in(Rp/pRp)[X], hence there exist polynomialsU1, U2, U3∈ Rp[X] such that 1=U1(X)Pi(X)+U2(X)T (X)+π U3(X). Now P (α)= 0= Π^rj=1P_jê^j(α)+π T (α), henceΠ^rj=1P_jê^j(α)= −π T (α), so

π=π U1(α)Pi(α)+π²U3(α)−Π^rj=1P_j^e^j(α)U2(α)

=π²U3(α)+Pi(α)U4(α) (2.8)

with U4=π U1−P_i^eⁱ⁻¹(r

j=1, j≠iP_j^e^j)U2 ∈Rp[X]. It follows fromLemma 2.6 thatᏮiᏭᏮi =Pi(α)ᏭᏮi, in other words,ᏮiᏭᏮi is principal. We conclude that ᏭᏮiis a DVR and therefore an integrally closed ring, and(OL)(p)=Rp[α].

(b) We will now prove that(P¯i,T )¯ =1 in(Rp/pRp)[X]for everyi=1, . . . , rif (OL)(p)=Rp[α]. We ﬁrst show that the ringᏭᏮiis a DVR, for everyi. Indeed, Rpis a Dedekind ring andLis a ﬁnite extension ofK, and it follows from [10, Theorem 6.1, page 23] that(OL)(p)=Rp[α]=Ais a Dedekind ring, soᏭᏮi is a DVR. Let us show next thatT (α)is a unit in everyᏭᏮi. Indeed,ᏭᏮiis a DVR and so its maximal idealᏮiᏭᏮi=πᏭᏮi+Pi(α)ᏭᏮi is principal. Letλ∈ᏭᏮi

be a generator of ᏮiᏭᏮi. Then there existu, v ∈ᏭᏮi such thatλ =π u+ Pi(α)v ∈ᏮiᏭ_Ꮾ_i−(ᏮiᏭ_Ꮾ_i)². NowRp is a DVR, P =Irrd(α, Rp), ¯P =Π^r_j=1P¯_j^e^j

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in(Rp/π Rp)[X],π Rp∈SpecRp, and(OL)(p)=Rp[α]=Ais the integral closure ofRpinL=K(α)withK=Fr(Rp), and we ﬁnd thatπ A=Π^rj=1Ꮾ^ej^j. Hence π∈Ꮾi2becauseei≥2. Nowλ ∈(ᏮiᏭᏮi)², hencePi(α)∉(ᏮiᏭᏮi)², because λ=uπ+Pi(α)v. It then follows thatPi(α)is a generator ofᏮiᏭᏮi=Pi(α)ᏭᏮi

since πᏭ_Ꮾ_i =(ᏮiᏭ_Ꮾ_i)^eⁱ =P_i^eⁱ(α)Ꮽ_Ꮾ_i, and π =Pie_i(α)1 with 1∈U (Ꮽ_Ꮾ_i).

We now show thatPj(α)∈U (ᏭᏮi)for everyj≠i. Indeed, ifPj(α)∈ᏮiᏭᏮi, then there exists ai ∈ Ꮾi and bi ∈ A−Ꮾi such that Pj(α)= ai/bi. Then ai=Pj(α)bi∈Ꮾi. Now,Ꮾi is a prime ideal of A, hencePj(α)∈Ꮾi. AsᏮj= π A+Pj(α)A, soᏮj⊆Ꮾi. The idealᏮjis a maximal ideal of A, soᏮi=Ꮾj. This is impossible because theᏮi are distinct, and it follows thatPj(α)∈U (ᏭᏮi) for everyj≠i. Thus there exists2∈U (ᏭᏮi)such thatr

j=1, j≠iP_j^e^j(α)=2. Sincer

j=1P_j^e^j(α)= −π T (α), π =P_i^eⁱ(α)1, and r

j=1, j≠iP_j^e^j(α)=2, then T (α)= −2₁⁻¹∈U (ᏭᏮi). SoT (α)∈U (ᏭᏮi)for everyi, and T (α)∈U (A);

otherwise, Krull’s theorem implies the existence of a maximal idealᏮi of A such thatT (α)∈Ꮾi, andT (α)∈ᏮiᏭᏮi=ᏭᏮi−U (ᏭᏮi), which is impossible.

We conclude thatT (α)is a unit inRp[α], and, by [2, Corollary 1, page 73], there existU1, V1∈Rp[X]such that 1=U1(X)P (X)+V1(X)T (X). Consequently ¯1= U¯1(X)P (X)¯ +V¯1(X)T (X)¯ in(Rp/π Rp)[X], which is principal. Hence(P ,¯T )¯ = 1 in (Rp/π Rp)[X] since ¯P = r

i=1P¯_i^eⁱ in (Rp/π Rp)[X] then (P¯i,T )¯ =1 in (Rp/π Rp)[X] for every i. Our result now follows fromProposition 2.2 and Lemma 2.7.

Remarks2.8. (1) Letπ be ramiﬁed inR[α],S = {i∈ {1, . . . , r} |ei≥2}, andf1(X)=

i∈SPi(X)∈R[X]. It follows fromLemma 2.7that the following statements are equivalent:

(i) (f¯1,T )¯ =1 in(R/p)[X];

(ii) vp(Res(f1, P ))=deg(f1);

(iii) for everyi∈S, we havevp(Res(Pi, P ))=deg(Pi), wherevpis thep-adic discrete valuation associated top.

(2) It follows from the above equivalence andRemark 2.4(2) and (3) that the condition inTheorem 2.5is independent of the choice of the monic lifting of ¯Pi. More precisely, ifei≥2 for everyi, and if we take another monic liftingQiof ¯Pi, thenvp(Res(Pi, P ))=deg(Pi)for alli=1, . . . , rif and only ifvp(Res(Qi, P ))= deg(Qi)for alli=1, . . . , r.

(3)Theorem 2.5states that, under the assumption thatei≥2 for everyi, OL=R[α]if and only ifR[α]isp-maximal for every prime idealpofR.

Corollary2.9. Under the assumptions ofTheorem 2.5, ifOL=R[α], then, for every prime idealpofR,Rp[α]is principal andᏮi=Pi(α)Rp[α]for everyi.

Proof. Indeed, a Dedekind ring having only a ﬁnite number of prime ideals is principal. To prove the second statement, takex∈Asuch thatᏮi=xA.

Then ᏮiᏭᏮi =xᏭᏮi =Pi(α)ᏭᏮi, hence Pi(α)=xε with ε∈ U (ᏭᏮi). Then ε∈U (A), soᏮi=Pi(α)A.

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Definition2.10. LetRbe a DVR with maximal idealm=π R, withf , g∈ R[X]monic polynomials. Thenf is called an Eisenstein polynomial relative togif there existsT∈R[X]and an integere≥1 such thatf=g^e+π T and (g,¯T )¯ =1 in(R/π R)[X].

Remark 2.11. As in the classical Eisenstein’s criterion, we have a criterion for the irreducibility of an Eisenstein polynomial relative to g, called the Schönemann criterion, see [12, page 273]; if f = g^e+π T is an Eisen- stein polynomial relative to g such that ¯g ∈ (R/m)[X] is irreducible and deg(T ) < edeg(g), thenfis irreducible inK[X].

Corollary2.12. LetRbe a DVR with maximal idealm=π R. IfP¯=g¯^ein (R/m)[X]withe≥2, thenOL=R[α]if and only ifPis an Eisenstein polynomial relative tog.

Proof. We obtain the result usingTheorem 2.5,Deﬁnition 2.10, andLemma 2.7.

Remark 2.13. Corollary 2.12 generalizes [14, Propositions 15 and 17]; it integrates the two results in one statement and provides the converse.

3. Monogenicity over the ring of integers. LetK=Q(α)be a number ﬁeld of degreen,P (X)∈Z[X]a minimal polynomial ofα,OK the ring of integers ofK, andpa prime number.

Proposition3.1. LetK=Q(α)be a number field andPthe minimal poly- nomial ofα. ThenOK=Z[α]if and only if for every prime numberpsuch that p²dividesDisc(P ), the prime numberpdoes not divideInd(α).

Proof. We obtain the result from the fact thatOK=Z[α]if and only if Ind(α)=1, and Disc(P )=(Ind(α))²dK(see [6], [4, page 166]).

Proposition3.2. LetP (X)¯ =r

i=1P¯_i^eⁱ(X)be the factorization ofP (X)mod- ulopinFp[X], and putf (X)=r

i=1Pi(X)withPi(X)∈Z[X]a monic lifting ofP¯i(X)andei≥2for alli. Leth(X)∈Z[X]be a monic lifting ofP (X)/¯ f (X)¯ andT (X)=(f (X)h(X)−P (X))/p∈Z[X]. Then the following statements are equivalent:

(i) pdoes not divideInd(α)=[OK:Z[α]];

(ii) (f ,¯T )¯ =1inFp[X];

(iii) vp(Res(f , P ))=deg(f );

(iv) vp(Res(Pi, P ))=deg(Pi), for everyi∈ {1, . . . , r}.

Proof. (i)(ii). Let(OK)(p)be the integral closure ofZ(p)inK. We ﬁrst show thatpdoes not divide Ind(α)if and only if(OK)(p)=Z(p)[α]. By the ﬁniteness theorem [13, page 48],(OK)(p)= ⊕ⁿ_i=0⁻¹Z(p)xi, and, becauseZ(p) is principal, αⁱ= ⁿj=⁻0¹aijxjwithaij∈Z(p), and therefore[(OK)(p):Z(p)[α]]= |det(aij)|.

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On the other hand, Ind(α)=[OK:Z[α]]=[(OK)(p):(Z[α])(p)]=[(OK)(p) : Z(p)[α]], hence(OK)(p)=Z(p)[α]if and only ifpdoes not divide Ind(α)if and only if Ind(α)∈ ∪(Z(p))=Z(p)−pZ(p). Hence by the proof ofTheorem 2.5,p does not divide Ind(α)if and only if(P¯i,T )¯ =1 inFp[X]for everyi=1,2, . . . , r (in other words, if and only if(f ,¯T )¯ =1 inFp[X]).

(ii)(iii). By [2, Corollary 2, page 73], (f ,¯T )¯ =1 in Fp[X] if and only if Res(f ,¯T )¯ =Res(f , T )≠¯0 inFpif and only if Res(f , T )∈Z−pZ. On the other hand,

Res(f , T )=(−1)^{deg(f )}

p^{deg(f )} Res(f , P ). (3.1) (ii)(iv). We have(f ,¯T )¯ =1 inFp[X]if and only if Res(f , T )∈Z−pZ. On the other hand, Res(f , T )=r

i=1Res(Pi, T )and

Res Pi, T

=(−1)^deg(Pⁱ⁾ p^deg(Pⁱ⁾ Res

Pi, P

. (3.2)

Theorem3.3. LetK=Q(α)be a number field of degreen,P (X)∈Z[X]

a monic minimal polynomial ofα, and OK the ring of integers ofK. Assume P (X)¯ =r

i=1P¯_i^eⁱ(X)inFp[X], for every prime numberpsuch thatp²divides Disc(P ), withPi(X)∈Z[X]a monic lifting ofP¯i(X)andei≥2for1≤i≤r. ThenOK=Z[α]if and only if for every prime numberp, such thatp²divides Disc(P ),vp(Res(Pi, P ))=deg(Pi)for1≤i≤r.

Proof. It suﬃces to apply Propositions3.1and3.2, andTheorem 2.5.

Remark3.4. Proposition 3.2provides a complement to the Dedekind criterion (see [4, page 305]). Indeed, inFp[X], we have(f ,¯T )¯ =(f ,¯T ,¯ h)¯ since all ei≥2.

We ﬁnish this section giving other conditions equivalent top not being a divisor of Ind(α).

Proposition3.5. The following statements are equivalent:

(i) pdoes not divideInd(α)=[OK:Z[α]];

(ii) Z[α]+pOK=OK; (iii) Z[α]∩pOK=pZ[α].

Proof. (ii)(iii). Consider the following map ofFp-vector spaces:

j:Z[α]/pZ[α] →OK/pOK, j

x+pZ[α]

=x+pOK. (3.3)

As OK and Z[α]are two free groups of the same rank n, Z[α]/pZ[α] and OK/pOK are two Fp-vector spaces of the same dimension n and injectivity ofjis equivalent to surjectivity ofj. Moreover,j is one-to-one if and only if Z[α]∩pOk=pZ[α]andjis onto if and only ifZ[α]+pOK=OK.

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(i)(iii). Ifpdoes not divide Ind(α)andpZ[α]⊂Z[α]∩pOK, then there existsx∈OKsuch thatx∉ Z[α]andpx∈Z[α], so the order of the subgroup generated byx+Z[α]of the ﬁnite groupOK/Z[α]is equal top, and, by La- grange’s theorem,pdivides Ind(α), which is the order of the groupOK/Z[α], and this is impossible.

Conversely, assume thatZ[α]∩pOK=pZ[α]andpdivides Ind(α). Cauchy’s theorem implies that there exists an element of orderpinOK/Z[α]; in other words, there exists x∈OK such thatx∉ Z[α] andpx∈Z[α]. Thenpx∈ Z[α]∩pOK=pZ[α], hencex∈Z[α], which is impossible.

4. Applications

4.1. Monogenicity of cyclotomic fields

Proposition4.1. Letn≥3be an integer,ξna primitiventh root of unity, K=Q(ξn), andφn(X)thenth cyclotomic polynomial overQ. ThenOK=Z[ξn].

Proof. We know from [15] that

φn(X)=

1≤i≤n i∧n=1

X−ξnⁱ

=Irrd ξn,Q

,

Disc φn

=(−1)^ϕ(n)/2 n^ϕ(n)

p|np^{ϕ(n)/(p−1)} =(−1)^ϕ(n)/2 s i=1

p^ϕ(n)(r_i ⁱ⁻^1/(pⁱ⁻¹⁾⁾, (4.1)

whereϕ(n)is the Eulerϕ-function and

n= s i=1

p_i^rⁱ=p^r_iⁱmi withmi= s j=1, j≠i

p^r_j^j. (4.2)

Letqbe a prime number such thatq²divides Disc(φn). Then there existsi∈ {1, . . . , s}such thatq=pi. We have ¯φn(X)=(φ¯m_i(X))^ϕ(p^riⁱ ⁾ (modpi), where ϕ(p_i^rⁱ)≥2, and

Res

φm_i, φn

=(−1)^ϕ(mⁱ^)ϕ(n)Res

φn, φm_i

=Res

φn, φm_i

=p_i^ϕ(mⁱ⁾, (4.3)

and we obtain thatvp_i(Res(φn, φm_i))=deg(φm_i(X)).

Now the result follows immediately fromTheorem 3.3andProposition 3.2.

4.2. Monogenicity of the fieldK=Q(α), withαa root ofP (X)=X^p−a Proposition4.2. Letαbe a root of the irreducible polynomialP (X)=X^p− a, whereais a squarefree integer andpis a prime number.

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(i)Ifpdividesa, thenOK=Z[α]if and only ifais squarefree.

(ii)Ifpdoes not dividea, thenOK=Z[α]if and only ifais squarefree and vp(a^p−1−1)=1.

Proof. We haveP (X)=X^p−a=Irrd(α,Q)and

Disc(P )=(−1)^p((p⁻^1)/2)NK/Q P(α)

=(−1)^(3p²⁻^p⁻^2)/2p(ap)^p⁻¹. (4.4)

Ifpis odd, the only prime numbersqsuch thatq²divides Disc(P )arepand the prime divisors ofa. Ifp=2, then 2 is the only prime numberqsuch that q²divides Disc(P ).

Letqbe a prime number such thatq²divides Disc(P ). We have two cases:

(1) ifqdoes not dividea, then ¯P (X)=g(X)^pinFp[X], withg(X)=X−a, and then Res(g, P )=P (a)=a^p−a;

(2) ifq dividesa, then ¯P (X)=g(X)^p in Fq[X], withg(X)=X and then Res(g, P )=P (0)= −a.

In both cases, the result is deduced fromTheorem 3.3.

References

[1] M. F. Atiyah and I. G. Macdonald,Introduction to Commutative Algebra, Addison- Wesley, Massachusetts, 1969.

[2] N. Bourbaki,Algèbre, Masson, Paris, 1981.

[3] M. Charkani,Structure multiplicative des idéaux d’une extension primitive d’un anneau intégralement clos, in preparation.

[4] H. Cohen,A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1996.

[5] R. Dedekind,Über den Zussamenhang zwischen der Theorie der Ideals und der Theorie der hoheren Cyclotimy index, Abh. Akad. Wiss. Göttingen Math.- Phys. Kl.23(1878), 1–23 (German).

[6] A. Fröhlich and M. J. Taylor,Algebraic Number Theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, Cambridge, 1993.

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[8] ,Z-bases d’entiers 1,θ,θ², θ³dans les extensions cycliques de degré4 deQ [Z-bases of integers1,θ,θ²,θ³in cyclic extensions of degree4of Q], Number Theory, Publ. Math. Fac. Sci. Besançon, Université de Franche- Comté, Besançon, 1981, pp. 1–14 (French).

[9] ,Non monogénéité de l’anneau des entiers des extensions cycliques deQ de degré premierl≥5 [Nonmonogeneity of the ring of integers of cyclic extensions ofQ of prime degreel≥5], J. Number Theory23(1986), no. 3, 347–353 (French).

[10] G. J. Janusz,Algebraic Number Fields, Pure and Applied Mathematics, Academic Press, New York, 1973.

[11] S. Lang,Algebra, 2nd ed., Addison-Wesley, Massachusetts, 1984.

[12] P. Ribenboim,Théorie des Valuations, Les presses de l’université de Montréal, Montréal, 1964 (French).

[13] P. Samuel,Théorie Algébrique des Nombres, Hermann, Paris, 1971 (French).

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[14] J.-P. Serre,Corps Locaux, Publications de l’Université de Nancago, Hermann, Paris, 1968 (French).

[15] L. C. Washington,Introduction to Cyclotomic Fields, Graduate Texts in Mathemat- ics, vol. 83, Springer-Verlag, New York, 1982.

M. E. Charkani: Department of Mathematics, Faculty of Sciences Dhar-Mahraz, Uni- versity of Sidi Mohammed Ben Abdellah, BP 1796, Fes, Morocco

E-mail address:[email protected]

O. Lahlou: Department of Mathematics, Faculty of Sciences Dhar-Mahraz, University of Sidi Mohammed Ben Abdellah, BP 1796, Fes, Morocco

E-mail address:[email protected]