Galois theory and integral models of Lambda-rings

0.16 MB·English

by  James Borger

GALOIS THEORY AND INTEGRAL MODELS OF Λ-RINGS
JAMESBORGER,BARTDESMIT
8
0
0 Abstract. We show that any Λ-ring, inthe sense of Riemann–Roch theory,
2 whichisfinite´etaleovertherationalnumbersandhasanintegralmodelasa
Λ-ringiscontainedinaproductofcyclotomicfields. Infact,weshowthatthe
n
categoryofthemisdescribedinaGalois-theoreticwayintermsofthemonoid
a
J of pro-finite integers under multiplication and the cyclotomic character. We
also study the maximality of these integral models and give a more precise,
5
integralversionoftheresultabove. Theseresultsrevealaninterestingrelation
1
between Λ-ringsandclassfieldtheory.

]
T
K
. Introduction
h
t According to the most common definition, a Λ-ring structure on a commutative
a
m ring R is a sequence of set maps λ ,λ ,... from R to itself that satisfy certain
1 2
compleximplicitly statedaxioms. This notionwasintroducedbyGrothendieck[5],
[
underthenamespecialλ-ring,togiveanabstractsettingforstudyingthestructure

1
onGrothendieck groupsinherited from exterior poweroperations; and as far as we
v
are aware, with just one exception [2], Λ-rings have been studied in the literature
2
5 for this purpose only.
3 However,it seems that the study ofabstractΛ-rings—thosehavingno apparent
2 relation to K-theory—will have something to say about number theory. One ex-
.
1 ample of such a relationship is the likely existence of strong arithmetic restrictions
0 on the complexity of finitely generated rings that admit a Λ-ring structure. The
8 primary purpose of this paper is to investigate this issue in the zero-dimensional
0
case. Precisely, what finite ´etale Λ-rings over Q are of the form Q⊗A, where A is
:
v a Λ-ring that is finite flat over Z?
Xi We will consider Λ-actions only on rings whose underlying abelian group is tor-
sion free, and giving a Λ-action on such a ring R is the same as giving commuting
r
a ring endomorphisms ψ : R → R, one for each prime p, lifting the Frobenius map
p
modulo p—that is, such that ψ (x)−xp ∈ pR for all x ∈ R. (The equivalence of
p
thiswithGrothendieck’soriginaldefinitionisprovedinWilkerson[8].) Anexample
that will be important here is Z[µ ] = Z[z]/(zr −1), where r is a positive integer
r
andψ sendsz tozp. Amorphismoftorsion-freeΛ-ringsisthesameasaringmap
p
f that satisfies f ◦ψ =ψ ◦f for all primes p.
p p
Note that if R is a Q-algebra,the congruence conditions in the definition above
disappear. Also, Galois theory as interpreted by Grothendieck gives an anti-
equivalence between the category of finite ´etale Q-algebras and the category of
finite discrete sets equipped with a continuous action of the absolute Galois group
G with respect to a fixed algebraic closure Q¯. Combining these two remarks, we
Q
Date:February2,2008. 18:01.
Mathematics Subject Classification: 13K05(primary);11R37,19L20,16W99(secondary).
1
2 J.BORGER,B.DESMIT
see that the category of Λ-rings that are finite ´etale over Q is nothing more than
thecategoryoffinite discretesetsequippedwithacontinuousactionofthe monoid
G ×N′, where N′ is the monoid{1,2,...} under multiplication with the discrete
Q
topology. This is because N′ is freely generated as a commutative monoid by the
prime numbers.
It is not always true, however, that such a Λ-ring K has an integral Λ-model,
by which we mean a sub-Λ-ring A, finite over Z, such that Q ⊗ A = K. In
order to formulate exactly when this happens, we write Zˆ◦ for the set of profinite
integers viewed as a topological monoid under multiplication, and we consider the
continuous monoid map
G ×N′−→Zˆ◦
Q
givenbythecyclotomiccharacterG →Zˆ∗ ⊂Zˆ◦ onthefirstfactorandthenatural
Q
inclusion on the second. Note that this map has a dense image. Now let K be a
finite´etalealgebraoverQ,andletS bethesetofringmapsfromK toQ¯. Suppose
K is endowed with a Λ-ring structure, so that we have an induced monoid map
G ×N′−→Map(S,S),
Q
where Map(S,S) is the monoid of set maps from S to itself. With the discrete
topology on Map(S,S) this map is continuous.
0.1. Theorem. The Λ-ring K has an integral Λ-model if and only if the action of
G ×N′ on S factors (necessarily uniquely) through Zˆ◦; in more precise terms, if
Q
and only if there is a continuous monoid map Zˆ◦ →Map(S,S) so that the diagram
GQ×NM′MMMMMMMMM&& //Zˆ(cid:15)(cid:15)◦
Map(S,S)
commutes.
It follows that the category of such Λ-rings is anti-equivalent to the category of
finite discrete sets with a continuous action of Zˆ◦ and that every such Λ-ring is
contained in a product of cyclotomic fields. It also suggests there is an interest-
ing theory of a Λ-algebraic fundamental monoid, analogous to that of the usual
algebraic fundamental group, but we will leave this for a later date.
Another consequence is that the elements −1,0 ∈ Zˆ◦ give an involution ψ
−1
(complex conjugation) and a idempotent endomorphism ψ on any K with an
0
integral Λ-model. In K-theory, the dual and rank also give such operators, but
here they come automatically from the Λ-ring structure. Also observe that any
element of the subset 0S ⊆S is Galois invariant and hence correspondsto a direct
factor Q of the algebra K. Therefore K cannot be a field unless K =Q.
In the first section, we give some basic facts and also show the sufficiency of
the condition in the theorem above. In the second section, we show the necessity.
The proof combines a simple applicationof the Kronecker–Webertheorem andthe
Chebotarev density theorem with some elementary but slightly intricate work on
actions of the monoid Zˆ◦.
The third section gives a proof of the following theorem:
0.2. Theorem. The Λ-ring Z[µ ] is the maximal integral Λ-model of Q⊗Z[µ ].
r r
GALOIS THEORY AND INTEGRAL MODELS OF Λ-RINGS 3
Ofcourse,the non-Λversionofthis statementis false—theusualmaximalorder
ofQ⊗Z[µ ]isaproductofringsofintegersincyclotomicfieldsandstrictlycontains
r
Z[µ ], if r >1.
r
A direct consequence of these theorems is:
0.3. Corollary. Every Λ-ring that has finite rank as an abelian group and has no
non-zero nilpotent elements is a sub-Λ-ring of a Λ-ring of the form Z[µ ]n.
r
We do not need to require that the ring be torsion free because any torsion
element in a Λ-ring is nilpotent, by an easy lemma attributed to G. Segal [3, p.
295]. We emphasize that while the definition of Λ-ring that we gave above does
notliterallyrequirethe ringtobe torsionfree,itisnotthe correctdefinitioninthe
absence of this assumption. In particular, Segal’s lemma and the corollary above
are false if the naive definition is used; for example, take a finite field. For the
definition of Λ-ring in the general case, see [5], [8], or [1].
Finally,manyofthequestionsansweredinthispaperhaveanaloguesovergeneral
number fields. There one woulduse Frobenius lifts modulo prime ideals ofthe ring
of integers,and then generalclass field theory and the class groupcome in, as well
asthetheoryofcomplexmultiplicationinparticularcases. Becausetheseanalogues
of Λ-rings are not objects of prior interest, we have not included anything about
them here. But it is clear that finite-rank Λ-rings, in this generalized sense or the
original,arefundamentallyobjectsofclassfieldtheoryandthattheyofferaslightly
differentperspective onthe subject. Itwouldbe interestingto explorethis further.
1. Basics
ThecategoryofΛ-ringshasalllimitsandcolimits,andtheyagree,asrings,with
those taken in the category of rings. (E.g. [1]) We will only need to take tensor
products,intersections,andimagesofmorphisms,anditisquiteeasytoshowtheir
existence on the subcategory of torsion-free Λ-rings using the equivalent definition
given in the introduction. For any ring R, let R[µ ] denote R[z]/(zr−1). Because
r
R[µ ]=R⊗Z[z]/(zr−1), the ring R[µ ] is naturally a Λ-ring if R is.
r r
Given a Λ-ring R, it will be convenient to call a Λ-ring K equipped with a map
R → K of Λ-rings an RΛ-ring. (Compare [1, 1.13].) When we say K is flat, or
´etale, or so on, we mean as R-algebras in the usual sense. We call a sub-Λ-ring of
a QΛ-ring a Λ-order if it is finite over Z. We do not require that it have full rank.
1.1. Proposition. Let K be a finite ´etale QΛ-ring. Then K has a Λ-order that
contains all others.
We call this Λ-order the maximal Λ-order of K.
Proof. BecauseanyΛ-orderAiscontainedintheusualmaximalorderofK,which
is finite over Z, it is enough to show that any two Λ-ordersA and B are contained
inathird. ButA⊗B isaΛ-ringthatisfinite overZ. SinceA⊗B isthecoproduct
inthecategoryofΛ-rings,themapA⊗B →K comingfromtheuniversalproperty
of coproducts (i.e., a⊗b 7→ ab) is a Λ-ring map. Therefore its image is a Λ-ring
that is finite over Z, is contained in K, and contains A and B. (cid:3)
1.2. Proposition. Let K ⊆L be an inclusion of finite´etale QΛ-rings. Let A⊆K
and B ⊆L be their maximal Λ-orders. Then A=K∩B.
4 J.BORGER,B.DESMIT
Proof. The intersection K ∩B is on the one hand a sub-Λ-ring of K and, on the
other, finite over Z. It is maximal among such rings because of the maximality
of B. (cid:3)
We can now prove the sufficiency of the conditions of theorem 0.1. For any ring
R,letR◦ denote Ritselfbut viewedonly asamonoidundermultiplication. Sothe
group R∗ of units is just the group of invertible elements of the monoid R◦.
1.3. Proposition. Let r be a positive integer, let S be a finite(Z/rZ)◦-set, and let
K be the corresponding finite ´etale QΛ-ring. Then K has an integral Λ-model.
Proof. TakeasetT (suchasS)admittingasurjection (Z/rZ)◦ →Sof(Z/rZ)◦-
`T
sets, the left side denoting the free (Z/rZ)◦-set generated by T. Let L be the
correspondingfinite´etaleQΛ-ring. Then K is naturally a sub-Λ-ringof L. On the
otherhandLis Q[µ ]T andsohasa Λ-modelZ[µ ]T. The intersectionofthis with
r r
K is then both a Λ-ring and an order of full rank in K. (cid:3)
2. Necessary conditions
Let K be a finite ´etale QΛ-ring admitting an integral Λ-model A, and let S =
Hom(K,Q¯) be the corresponding G ×N′-set.
Q
The purpose of this section is to show that there is an integer r > 0 such that
this action factors through the map G ×N′ → (Z/rZ)◦ given by the cyclotomic
Q
character on the first factor and reduction modulo r on the second.
As usual, we say a prime number p is unramified in A if A/pA has no non-zero
nilpotentelements. AprimeisramifiedinAifandonlyifitdividesthediscriminant
of A. Therefore the set of primes that ramify in A is finite and contains the set of
primes that ramify in the usual maximal order of K.
2.1. Proposition. Theendomorphism ψ ofAisanautomorphism ifandonlyifp
p
isunramifiedinA. Inthiscase,ψ istheuniqueliftoftheFrobeniusendomorphism
p
of A/pA.
Proof. If ψ is an automorphism, then the Frobenius endomorphism x 7→ xp of
p
A/pA is an automorphism, and so p is unramified.
Suppose instead that p is unramified. Then A/pA is a finite product of finite
fields, and so the Frobenius endomorphism of A/pA is an automorphism of finite
order. Thecategoryoffinite´etaleZ -algebrasisequivalenttothecategoryoffinite
p
´etale F -algebras, by way of the functor F ⊗ −. (See [4, IV (18.3.3)], say).
p p Zp
Thus the endomorphism 1⊗ψ of Z ⊗A is the unique Frobenius lift and it is an
p p
automorphism of finite order. It follows that ψ : A → A is the unique Frobenius
p
lift to A, and is also an automorphism. (cid:3)
2.2. Proposition. Thereisapositiveintegerc,divisibleonlybyprimesthatramify
in A, such that the action of G on S factors through the cyclotomic character
Q
G →(Z/cZ)∗. If p is unramified in A, then p∈N′ and (pmodc)∈(Z/cZ)∗ act
Q
in the same way on S.
Proof. Define the number field N to be the invariantfield of the kernel of the map
G →Map(S,S). Write G¯ =Gal(N/Q) and let O be the ring of integers of N.
Q N
Takeanyelementg ∈G¯. ByChebotarev’stheorem[7,V.6]thereisanunramified
primepofN lyingoveraprimenumberpsuchthatg(x)≡xp modpforallx∈O ,
N
i.e., g is the Frobenius element of p in the extension Q ⊂ N. Since Chebotarev’s
GALOIS THEORY AND INTEGRAL MODELS OF Λ-RINGS 5
theorem provides infinitely many such p, we may also assume that A is unramified
at p.
Wenowclaimthatforalls∈S =Hom(A,O )themapss◦ψ andg◦sfromAto
N p
O are equal. Since A is unramified at p, the map Hom(A,O )→Hom(A,O /p)
N N N
isinjective,soitsufficestoshowthattheircompositionswiththemapO →O /p
N N
are equal. But this follows from 2.1 and our choice of p. Thus, g ∈ G¯ and p ∈ N′
act in the same way on S.
It follows that the image of G in Map(S,S) is contained in the image of N′,
Q
so G¯ is abelian. By the Kronecker-Weber theorem [7, III.3.8], N is contained in
a cyclotomic field Q(µ ), where c is divisible only by primes that ramify in N.
c
Since N is the common Galois closure of the components of A⊗Q, such primes
are ramified in A as well. The last statement follows from the fact that for any
prime number p∤c the element (pmodc)∈(Z/cZ)∗ corresponds to the Frobenius
element of any prime over p in the extension Q⊂Q(µ ). (cid:3)
c
It follows that our map of topological monoids G ×N′ → Map(S,S) factors
Q
throughZˆ∗×N′. WewillshowthatitfactorsfurtherthroughZˆ◦ withthefollowing
criterion.
2.3. Proposition. AcontinuousactionofZˆ∗×N′ onafinitediscretesetT factors
through a continuous action of Zˆ◦ if and only if
(i) all but finitely many primes p∈N′ act as automorphisms on T, and
(ii) for all d ∈ N′ there exists an integer c such that the action of Zˆ∗ on dT
d
factors through (Z/c Z)∗ and for each n∈N′ with ndT =dT we have
d
— n is relatively prime to c , and
d
— the elements (nmodc ) ∈ (Z/c Z)∗ and n ∈ N′ act on dT in the
d d
same way.
Proof. To show the necessity of (i) and (ii), assume that the action of Zˆ∗ ×N′
factors through (Z/rZ)◦ for some integer r > 0. Then all primes not dividing r,
when viewed as elements of N′, act as automorphisms on T. This establishes (i).
Toshow(ii), takeanyd∈N′ andletc bethe smallestpositiveintegerc forwhich
d
the Zˆ∗-action on dT factors through (Z/cZ)∗. Note that c divides all c with this
d
property. NowsupposethatpisaprimewithpdT =dT. Writer =pne,withp∤e.
Thenfor anyx,y ∈(Z/rZ)◦ with x≡y mode andany s∈dT (in factany s∈T),
we have
pn xs =(pnx)s=(pny)s=pn(ys).
(cid:0) (cid:1)
Sincepactsbijectively ondT,thisimpliesthatxandy actinthe samewayondT,
so the action of Zˆ◦ on dT factors through (Z/eZ)◦. In particular, c | e, so p ∤ c ,
d d
and the elements p ∈ N′ and (pmode) ∈ (Z/eZ)∗ and (pmodc ) ∈ (Z/c Z)∗ all
d d
act in the same way on dT. Since N′ is generated by the primes, part (ii) follows.
For the converse,suppose (i) and (ii) hold. For every prime number p, let a be
p
the smallest integer a ≥ 0 such that paT = pa+1T. By (i) we have a = 0 for all
p
butfinitelymanyp,sor0 =Qppap isaninteger. Notethatforanyn∈N′ wehave
nT =gcd(n,r )T.
0
Now let r be any integer divisible by dc for every d|r . We will show that the
d 0
action of Zˆ∗×N′ on T factors through (Z/rZ)◦. To do this, we will show directly
that any two elements (a ,d ),(a ,d )∈Zˆ∗×N′ satisfying a d ≡a d modr act
1 1 2 2 1 1 2 2
in the same way on T.
6 J.BORGER,B.DESMIT
Sincer |rthecongruenceimpliesthatd andd havethesamegreatestcommon
0 1 2
divisor d with r , so we have d T = dT = d T. For i = 1,2 we define d′ ∈ N′
0 1 2 i
by d = dd′ and deduce that d′(dT) = dT. Using (ii) one sees that d′ is coprime
i i i i
to c , and that the action of d′ on dT is that of (d′ modc ) ∈ (Z/c Z)∗. By the
d i i d d
defining property of r we have dc | r, so a d′d ≡ a d′dmoddc , which implies
d 1 1 2 2 d
a d′ ≡ a d′ modc . It follows that (a ,d′) and (a ,d′) are mapped to the same
1 1 2 2 d 1 1 2 2
element of (Z/c Z)◦, whichin fact lies in (Z/c Z)∗. Thus, (a ,d′) and (a ,d′) act
d d 1 1 2 2
identicallyondT,andcomposingwith(1,d)weseethat(a ,d )and(a ,d )actin
1 1 2 2
the same way on T. (cid:3)
In order to finish the proof of theorem 0.1 one checks the conditions of 2.3 for
T =S. Condition(i)followsfrom2.1andthefactthatAisramifiedatonlyfinitely
many primes. For condition (ii), suppose d ∈ N′ is given and consider the sub-Λ-
ring ψ (A) of A which corresponds to the G ×N′-set dS of S. Proposition 2.2
d Q
applied to ψ (A) now provides an integer c so that the G -action on dS factors
d d Q
through (Z/c Z)∗. Any n ∈ N′ with ndS = dS is a product of primes that are
d
unramified in ψ (A) by 2.1, and so 2.2 tells us that (n mod c ) ∈ (Z/c Z)∗ and
d d d
that this element acts on dS in the same way as n. This gives condition (ii).
3. Explicit maximal Λ-orders
Given a prime number p, there is a notion of Λ -action on a ring R, and as
p
before,thishasasimple descriptionifRhasnop-torsion: aringendomorphismψ
p
ofR that lifts the Frobenius endomorphism, that is, suchthat ψ (x)−xp ∈pR for
p
all x ∈ R. Also as before, a sub-Λ -ring A of a Q Λ -ring K is called a Λ -order
p p p p
if it is finite over Z . It is said to be maximal if it contains every other Λ -order
p p
in K.
We have two natural ways of making Λ -orders. First, for any abelian group V
p
the groupringZ [V]is a Λ -ringwhenwe setψ (r)=r for r ∈Z andψ (v)=vp
p p p p p
for v ∈ V. Secondly, if A is the ring of integers of a finite unramified extension K
of Q , then A has a unique Λ -ring structure, where ψ is the Frobenius map (cf.
p p p
2.1). By extending ψ to K we see that A is the maximal Λ -order of K. Taking
p p
tensor products of these two building blocks we see that for any integer q the ring
A[µ ]=A[z]/(zq−1) is a Λ -order in K[µ ].
q p q
3.1. Lemma. If q is a power of p, then A[µ ] is the maximal Λ -order of K[µ ].
q p q
Proof. Byinduction,itisenoughtoassumeA[µ ]ismaximalandthenproveA[µ ]
q pq
is.
Let k denote the residue field of A, and let ζ denote a primitive pq-th root of
unity in some extension of K. Then we have K[z]/(zpq−1)=K(ζ)×K[y]/(yq−
1), the element z corresponding to (ζ,y). In these terms, the Λ -action is given
p
by ψ (b,f(y)) = (f∗(ζp),f∗(yp)), where f∗(y) denotes polynomial obtained by
p
applying the Frobenius map ψ coefficient-wise to f(y).
p
Now consider the following diagram of rings:
A[z]/(zpq−1) z7→y ////A[y]/(yq−1)
z7→ζ y7→y
(cid:15)(cid:15)(cid:15)(cid:15) (cid:15)(cid:15)(cid:15)(cid:15)
A[ζ] ζ7→y //// k[y]/(yq−1),
GALOIS THEORY AND INTEGRAL MODELS OF Λ-RINGS 7
whereA[ζ] denotesthe ringofintegersinK(ζ). As notedforexample inKervaire–
Murthy [6], this is a pull-back diagram. This is just an instance of the easy fact
that for ideals I and J in any ring R, we have
R/(I∩J)=R/I× R/J.
R/(I+J)
In our case, take R=A[z], I =(zq−1), and J =(1+zq+···+zq(p−1)).
Now suppose R is a Λ -order in K[µ ]. Then the image of R in K[y]/(yq−1)
p pq
is contained, by induction, in A[y]/(yq −1). Therefore R is contained in A[ζ]×
A[y]/(yq−1);wewillviewelementsofRaselementsofthisproductwithoutfurther
comment. Because the diagram above is a pull-back diagram, we need only show
thatthetwomapsR⇉k[y]/(yq−1)givenbymappingtothetwofactorsandthen
projecting to k[y]/(yq−1) agree.
LetvdenotethevaluationonA[ζ]normalizedsothatv(p)=1. Leta61/(p−1)
be the largest number such that
(3.1.1) for all (b,f(y))∈R we have v(b−f(ζ))>a.
Notethattheexpressionf(ζ)makessenseonlymoduloζ −1,andsothecondition
p
above is meaningless if a>v(ζ −1)=1/(p−1).
p
Let(b,f(y))beanelementofR. ThenbecauseRisaΛ -ring,thereisanelement
p
(c,g(y))∈pR such that
(c,g(y))=(b,f(y))p−ψ (b,f(y))=(bp−f∗(ζp),f(y)p−f∗(yp)).
p
On the other hand, because of our assumption on a, we have
1+a6v(c−g(ζ))=v((bp−f∗(ζp))−(f(ζ)p−f∗(ζp))=v(bp−f(ζ)p).
But because the integral polynomial p(X −Y) divides (X −Y)p−(Xp−Yp), we
have
v((b−f(ζ))p−(bp−f(ζ)p))>1+a
and hence v(b−f(ζ))>(1+a)/p. That is, (1+a)/p satisfies (3.1.1).
But then a = 1/(p−1) because otherwise this would violate the maximality
of a. In other words, for any element (b,f(y)) ∈ R, the element ζ −1 divides
p
b−f(ζ). This is just another way of saying b and f(y) have the same image in
A[ζ]/(ζ − 1) = k[y]/(yq − 1), and hence the element (b,f(y)) lies in the fiber
p
product, which we showed above is A[µ ]. (cid:3)
pq
3.2. Remark. It is not true that Z [V] is maximal for every finite abelian group
p
V. For example, if V = Z/pZ×Z/pZ and x = 1 σ, then ψ (x) = p and
pPσ∈V p
x2 = px. Therefore, the subring Z [V][x] of Q [V] is a sub-Λ -ring which is finite
p p p
overZ . The globalanaloguealsoholds: Z[V]isnotthe maximalintegralΛ-model
p
for Q[V]. This follows from the local statement.
3.3. Proposition. Let K be a finite ´etale QΛ-ring, and let R be a Λ-order. If
Z ⊗R is a maximal Λ -order in Q ⊗R for all primes p, then R is maximal.
p p p
Proof. Let S be the maximal Λ-order in Q⊗R. Because Z ⊗R is the maximal
p
Λ -orderinQ ⊗R,the inclusionZ ⊗R⊆Z ⊗S isanequality. ThereforeR and
p p p p
S have the same rank, and p does not divide the index of R in S. (cid:3)
3.4. Theorem. Let r > 1 be an integer. Then Z[µ ] is the maximal Λ-order in
r
Q[µ ].
r
8 J.BORGER,B.DESMIT
Proof. By 3.3, it suffices to show that for every prime p, the Λ -order Z [µ ]
p p r
maximal. Write r = qn, where q is the largest power of p dividing r. Then
Z [µ ] = Z [µ ][µ ] = A[µ ], where A runs over the irreducible factors of
p r p n q QA q
Z [µ ], all of which are unramified extensions of Z . By 3.1, each factor A[µ ] is a
p n p q
maximal Λ -order, and therefore so is their product. (cid:3)
p
3.5. Remark. Using 0.3 and 1.2, we can also describe the maximal Λ-order in
generalasfollows. LetS be afinite (Z/rZ)◦-set,letζ ∈Q¯ denote aprimitive r-th
r
root of unity, and let K = Hom(Z/rZ)∗(S,Q(ζr)) denote the corresponding finite
´etale Λ-ring over Q. Consider the isomorphism
Q[µr]→YQ(ζrd)
d|r
givenbyz 7→(...,ζd,...) . ThenthemaximalΛ-orderinK isthesetofelements
r d|r
f ∈K such that for all s∈S, the element
(...,f(ds),...)d|r ∈YQ(ζrd)∼=Q[µr]
d|r
lies in Z[µ ].
r
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James Borger, Mathematical Sciences Institute, Building 27, Australian National
University,ACT 0200,Australia
Bart de Smit, Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA
Leiden, TheNetherlands
E-mail address: [email protected], [email protected]