ciclotomic fields with unic factorizationrll.1976.286-287

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Cyclotomic fields with unique factorizationB y /.Myron Masley* at Chicago and Hugh L. Montgomery* at Ann Arbor

1. Statement of results2m

For a natural number w>2, we le t Cm = Q(em) be the ra-th cyclotornic field,of degree (m) over the field Q of rational numbers, we let h m denote the class numberof C m , and w e let/? be a prime num ber. l f m is odd then Cm = C2m, so to avoid confusion wesuppose always that m 2 (mod 4). Kummer conjectured that the integers in C p haveunique factorization (that is , hp= 1) if and only if p :g 19. This conjecture was in principleestablished by Ankeny and Chowla [1], R. Brauer [4], Tatuzawa [17], and later bySiegel [16], each of whom proved an assertion which implies that hp— > oo. Here wemodify the analytic approach to obtain good numerical results, and we introduce algebraicconsiderations to prove th e following

Main Theorem. There are precisely twenty-nine distinct cyclotornic fields Cm withclass number h m = 1. They are given by m = 3, 4, 5,7,8,9, 11, 12, 13, 15, 16, 17, 19 ,20 ,21 ,24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 48, 60, 84.

One should keep our initial remark in mind, äs there are fifteen additional m withm =2 (mod 4) such that h m = l; the fields Cm which arise in this way are listed aboveäs G! .

im

One m ay note from the Main Theorem that if h m = l then (m ) <; 24. We also observethat the above gives the following

Corollary (Kummer's Conjecture). The prime cyclotornic field C p has class numberh p = l ifand only ifp ^19.

Uchida [18] has independently proved this corollary.

Before describing our line of attack we recall a few Standard facts concerning

cyclotomic fields. The field C+ = Q l cos ), with class number A + , is the maximal real

V

m

)subfield of C m . Moreover, h* divides h m so we may write h m = h*h^9 where A* is the relativeclass number, the so-called first factor of hm. In general the second factor is the harder oneto deal with, äs expressions for h+ usually involve th e units in C + .

* Portions of this research were undertaken while the first author was an NSF fellow at PrincetonUniversity, and w hile the second author was a Marshall Scholar at Trinity College, Cam bridge .



Masley a nd Montgomery, Cych tomic ficlds \vith uniquefactorizalion 24 9

In proving the Main Theorem, our first Step is to establish a lower bound for A * , whichwe use to show that h* = l if and only i f p ^ 19. We then show that if m \n then A * | A * so thatif A * = l then m is composed of primes p ^19. For these primes we determine the highestpower /?a of p for which A * « = l. We thus obtain a fmite list of possible m; we shortenthis list by showing that if m has four or more distinct prime factors, ù(m) ^4, then 2|A*.

Then a short calculation enables us to determine the twe nty-nine field s for which A* = l.Finally we show that h* = l when A* = l , so we have our Main Theorem.

We have been pleased to receive a number of comments and suggestions regardingthis work. In particular we are happy to thank Professors Davenport, Iwasawa,Mets nkyl , M . Newman, A. Seiberg, and K. Uchida for their assistance.

2. Analytic Lemmas

We assume throughout this section that p > 200, we let

(2.1) /(*)= Ó

where the sum is over all characters ÷ (modp) with ÷( — 1) = — l, and we write s =ó + / / .Clearly f(s) is multiple- valued, but from known zero-free regions (see Estermann [6],

Theorems 39, 40, 46, and Davenport [5], p. 99 with c21 =-7-) we know that the L (s, ÷)6

do not vanish for

(2 .2) |s-l|:g2/>-3.

Thus /(s) is single-valued in the union of this disc and the half-plane ó> 1. Our goal is toestimate/(l). To this end we establish the following lemmas.

Lemma 1. 1/ó >l then |/(ó)| ^3 + log P

ó — i

Proof. W e write/(a) s a Dirichlet series

,=! log« V 2 / » = i log«

say. Then — T 2 /̂( " ) < 7i. We bound T 2 and note that our argument may be modifiedto provide the same bound for Ji.

Ifx>pthen(Montgomery and Vaughan [13], Theorem2)2x

é ×( p -1) log ( —

so



250 Masley and Montgomery, Cyclotomicfieldswithuniquefactorization

_w .f â w < f2 W 2 W- -

Let

/ \ ^-~*

V 5:/ / ^̂

njZx log«

It is easily seen that

4-22^k^21ogjc \P / /^

from which it follows that

y2 „ l o g « 2oo 7

</? J (2yxp~l+logx)x~2dx< — 4 - log/?.p 5

Lemma 2. Lei ó0 = l + / > ~ 3 . TAe« Re/(5) ^— / ? \ogpfor

(2.3) |*-ó

. B y partial summation

for ó>0, where 5(w) = Ó *(«)· B ut |S(w)| ^—p, so

in the disc under consideration. Hence Re log L ( s 9 x)<logp which gives the result.

Lemma 3. With ó0 s above, if\ ^ó^ó0 i A e w |/'(ó)|^4/;4 log/?.



Masley and Montgomery, Cyclotomic fields with unique factorization 251

Proof. The disc (2. 3) lies in the union of the disc (2. 2) and the half-plane ó> l,so g(s) =f(s) — /(ó0) is regul r for s satisfying (2.3). From Lemma 2 we see thatReg(s)^p log/? when (2.3) holds. The Borel-Caratheodory lemma (Estermann [6],Theorem 7) implies that if A (z) is regul r in a domain containing the disc |z — a\ ^R,

O T L f 4

if A(a) =0, and if R e A ( z ) ^ M in the disc, then \h'(z)\^—— in |z-a|£— R. Applied

R 2) > this gives |g'(s)| ^4p4 log/? for |s — ó0| ^ / > ~ 3 . B ut /'(s) =g'(^), so we are done.

3. Kummer's conjecture

We now use the formula (cf.Hasse [8],§§ 2—5)

p+3 p-3 p-1

(3.1) A * =p * 2" ^ „- 2 n

÷ ( - À ) = - é

to estimate h*. From this formula and the lemmas of the previous section we derive

Theorem 1. I f p > 200 then

where -l ^0^1.

In fact it is known that 0 = 0(1) in the above s / ? — oo, but the proof of this is

non-effective. In an earlier version of this paper we established a relation som ewhat weake rthan the above, using a substantially different method. We do not describe our earlierargument here, but it may be reconstructed from the work of Mets nkyl [12], whosubsequently used the method to show that if p* + 1 > 100,á ^l, then h *« + 1 > A *« .

Proof. In view of (3. 1), we have only to show that

yIog2 +y1ï§ð+ |/(1)| ^7 logp ,

where/(5) is given by (2. 1). B ut/(!)=/(*)-/'(£)(*-!)

for some î, l < î <ó, so that if ó^ó0 then

|^3 +log-—+4(ó-1)/?4 log/7 ," — l

by Lemmas l and 3. We take ó= l - f / ?~ 5 , and the desired result follows, since

3 1 4 loe/7— Iog2-l· — É ï % ð +3 4--—<log/72 2 /?

for/»200.

Corollary 1. PFe Aat;^ A* = l if and only ifp^l9. Moreover, A * ^ 1020 if and only if



252 Masley and Montgomery, Cyclotomicf ields with uniquefactorization

Proof. The least prime p > 200 is p = 211, and from Theorem l we see that A* > 1020

f o r / 7 ^ 2 1 1 . The numbers A£ , 3^/7^200 have been calculated independently by Schrutka[15] and M. Newman [14],and in shorter ranges by K umme r and Hasse [8]. (It has beenfound that three values given by K umme r are incorrect; see [14].) From these tables we findthat A * ^ 1020 precisely w h e n /? _^ 131. If we were to use the f ll extent of Schrutka's longertable we would find that A* < 1035 precisely when/7 ^181.

4. Divisibility properties of h %

In this section we establish some divisibility properties of A* and use them todetermine those m for which A* = l .

W e require a formula for A* which may be described s follows (cf. Hasse [8],pp. 78 — 79). Let £be the group of characters modm. For each cyclic subgrou p of£choosea generator ÷, le t ø (mod/^) be the primitive character which induces ÷, and let çø beth e order of ø. Let Ø be the set of such ^ with ø( — !)=—!. Let Íö be the norm mapÍø : Cni/r*0, and put

e* = - Ó áø(á).

Then

(4-1) h* =Q w Ð ÍÖ(ÈØ)é / / å Ø

where Q= l if m is a prime power and 0= 2 otherwise, and w is the number of rootsof unity in C m , w =2m if m is odd, w = m if m is even. We need the following properties

Lemma 4. Let ö be a character m o d / ^ with ö(—\)= — l and let çö be the order of\\i.

A) If/ø is divisible by m o r e than one prime then Íø(Èø) e Z.

B ) Suppose n^ is a power of2.

i) Ifat least three distinct primes dividef^ then ÍÖ(ÈÖ) 6 2 Z .

ii) If only tw o dist inct prim es p,q,p^2 divide /^,, then Íö(Èø) is an even or odd

f q \integer according s th e quadratic residue Sym bol — = l or — l.\P )

Q If f i i /=P * > > then Íø(Èø)€-Z and, m o r e precisely , the denominator (in lowest

terms) ^

i) may bedivisible bypi

n) is ±4forft =4and± 2forf^ =2á, á=3 ;

iii) is ±2 ifn^ = T (since \l/(-i)=-l,this occurs only fo r 2u\\p-1);

iv) is ± l in al l other cases.



Masley and Montgomery^ Cyclotomic fields with unique factorization 253

Proof. This is the content of S tze 30—33 in Hasse [8], §§ 30—32. We now use (4. 1)to prove the following tw o lemmas :

Lemma 5. Ifm\n then A * | A * .

Lemma 6. Iffour distinct primes divide m, then 4|A* .

Proof of L e mma 5. It suffices to prove that for ( p , ra) = l we have A * | A * p and^ m p « l ^ m p « + 1 for á ^l. Let A £ = ( w)b nb be a shorthand notation for (4. l) . Then

(4 2) h** = ^w^pn^p '

If m is a prime power, (4. 2) becomes 2p Ð ^((9^) e Z by Lemma 4. If at least tw op \ f *

distinct primes q and r divide m, we get a character ^ of type B ) i) in Lemma 4. Such a ^is given for example by ø(æ) = îñ(æ ) ! ;q(z) î,(æ) for (z ,/?#r) = l where î2 is the quadratic

f-l

character m o d 4 an d îß = ÁÃ

ß

2

" with X t a generator of the group of characters m o d f ,2"| 1 1 — l if / is an odd prime. Now (4. 2) becomes

"»p —n n mp

~h*~~~ P P~JT1Tlm 11 „ 11n

n,v

=p ( element of — — Z l (element of 2Z) e Z.V

2P J

Now consider

(4.3) /7 mpa

For/? odd or á > l, (4. 3) becomes

— P Ð Íø (Èø) =ñ ( element of — Z l e Z by Lemma 4.Pa + 1 I A \ P /

If j r ? a = 2 and m is a prime power, (4. 3) becomes 4 l element of — Z 1 . If p* = 2 and m is not

a prime power, we get a ^ of type B) i) s described befo re and (4. 3) becomes

=2(̂ P| Ð (̂0̂ )=\4 / 4 L / V ,

again by Lemma 4. We note that it is known that m \n implies A ^ | A ^ " and also A m | A „( c f . Ankeny, Hasse, Chowla [2], Theorem 3).

Proof of L e mma 6. By Lemma 5 it suffices to show tha t h* qrs and h% pqr are divisible by 4for distinct odd primes /?, q, r, and s. The product of the non-integral factors in /7m form=pqrs or m = 4pqr has denominator (in lowest terms) at most 8w by Lemma 4. The

factor w is canceled automatically in A * = ( v t > ) m / 7 m . Now for each divisor o f m whichconsists of exactly three primes a, b , c we have the character ø =444of type B) i) (see

/4\proof of Lemma 5 above). Hence, the integral part of T I m has a t least 1 1 = 4 factors of 2.

Since â = 2 in this case, A* = ( w)m U m is divisible by 4.

J o ur nal f r Mathem atik. Band 286/287 33



254 Masley and Montgomery, Cyclotomic fields with uniquefaciorization

We now determine when A* = l .

Proof. By Lemma 5 and Corollary l we may restrict our attention to p ^19.In the tables of A* in Schrutka [15], we find that h* = l for the values of m listed aboveand that h* > l for m = 26, 34, 53, 72, II2, and 132. To complete the proof it suffices to showthat A f 7 2 and /zf 92 are not 1. In Masley [11] it is shown that integral factors of Af ? 2 andA * 9 2 are Ö l modq for certain primes q. Hence Af ? 2 and /zf 92 cannot be 1. An independentproof ofthis is provided by the result of Mets nkyl mentioned above.

Theorem2. Suppose that ra ö 2 (mod4). Then h* = l i fand only ifm = 3, 4, 5, 7, 8, 9,11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 48, 60, or 84.

Proof. Corollary l, Lemma 5, and Lemma 6 show that if A* = l, then m=paqbrc

where/?, q,and r are primes ^19 and a, fc, c are non-negative integers. Lemma 5 and Lemma 7put further restrictions on a, ft, and c. We observe that A * 7.1 9 is even by Lemma 4, B)ii),

for example. Other possible cases for m =paqbrc to satisfy A* = l are handled by checkingthe tables of Schrutka [15] or by using Lemma 4.

5. Determinationof h +

In this section we determine those cyclotomic fields Cm with class number one byshowing that h„= l wheneverA* = l. We shall use the following two lemmas.

Lemma 8. Lei K be a finite algebraic n u mb e r f ie ld and L/K a Galois extension ofdegree a power of p. Suppose that exactly o ne prime of K is ramified in L. Then p\hL only ifp\hK, where h K andhL are the class num bers o fKand L, respectively.

This is the content of Theorems 3 and 4 of Yokoyama [19]. Iwasawa [9] establishedthe special case of Lemma 8 in which L/K is a cyclic extension and the ramified prime isfully ramified.

Lemma 9. Leip be an odd prime number, L/K anabelian extension oftype (2, 2) and hL

and hK the class n u mb e r s of L and K respect ively . Then p\hL implies that p divides th e classn u mb e r ofat least oneofthe threeproper intermediatef ie lds between L and K.

Proof . Let (7 = {l, ó, ô, óô} be the Galois group of L/K and let Lt be the fixed fieldof {l, t} for t =ó, ô, óô. For a field F let S F be the /?-Sylow subgroup of CF, the ideal classgroup of F.

Assume first that p\hK. We shall show that then p\hLt for ß = ó, ô, óô. If p)(hLt thenthe class of some -ideal á has order p in C K but is principal s an Lrideal. Since Lt/K isof degree 2, taking ideal norms shows that a2 is equivalent to a principal T-ideal; that is,the class of á has order 2 in CK. However, the class of á was assumed to have order/?.

Hence we may assume that p)(hK. G acts on CL and on SL in the natural way.SL isuniquely divisible by 2 since it is abelian of odd order. Consequently we may extend the action

of G on SL linearly to an action of D = (I — l [G] on SL. We write this action expo-' " - m+ó 1 ó

nentially. For example, for a e SL, a 2 =a2a2 where a2 is the solution of x2 = a° in SL.



Masley and Montgomery, Cyclotomicfieldswithuniquefactorization 255

Let 2å* = l ± t for t =ó, ô, óô. Then å* e D. For a subgroup // of SL, á e £ > , letH*={he H\h* = h} be the subgroup of H fixed by a. Since l = å ^ " + å ~ is a decompositionof l into orthogonal idempotents, S L = S * £ ÷ S ? £ is a direct product decomposition of SL.In a similar manner, using l = å ô

+ -f å ô ~ , we decompose the subgroups S * £ and Sjf and get

( 4 . 4 )

where, for example,restricts to

= {be S L\b B* =b*r~ = b}. The norm map of CL to CLa, say,

and this restriction is a monomorphism becauseand b2 in the principal class of LG implies that b2 is the identi ty in S £'orde r. In a similar fashion we obtain e mbedd ings

b 2 for beS£*~, a group of odd

where we observe that å ~ å ô ~ = å + ô å ~ = å + ô å ~ . The group CK has order H K prime to p byassumption so 5£·+ must be trivial. B y hypothesis, SL is non-trivial so by (4. 4) at leastone of S £ ' ~ , 5£f + , and S£'~ is non-trivial. This provides an element of order p in atleast one of C Ltr , CLr, or CL<rr so p divides the class numbe r of at least one of L G , L T, or LOT.

TheoremS. Suppose that m ö 2 (mod4). Then C m has class number one if and onlyi f ö (m) ^24, and m Ö 23, 39, 52, 56, or 72. Alternatively, C m has class number one if and

only ifm = 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 40,44, 45, 48, 60, or 84.

Proof. Since hm = h%h* we may restrict to those m for which A* = l. These areprecisely the m mentioned above, so we have only to show that h^ = l for these m .

When C^ is cyclic we find that h+ = l from computations of Bauer [3] which dependon Leopoldt's/7-adic class numb er formula [10].

It remains only to consider h* for m = 24, 40, 48, 60, and 84. We first show that eachof these class numbe rs has no odd prime divisor.

Since C8+ = Q ( ] 2 ) , C1+2 = Q ( ] 3 ) , a n d Q ( ] 6 ) haveclass numbe r one,no odd prime divides A ^ ~ 4 by Lemma 9.Similarly C^6 and F4S (the real cyclic field ofconductor 48) have class number one ([3] or [7]), soagain Lemma 9 shows that no odd prime divides h^s.Similar considerations show that A^,, h£0, and h£4 haveno od d prime divisor.

We now show that 2 / f A + for m =24, 40, 48, 60, and 84. For each of these m le t m0

be the largest odd divisor of m . Then CJCmo and C+/C* 0 are Galois extensions whosedegrees are powers of 2. For each of the five values of m we also know that hm o =h* 0.

For m = 24, 40, or 48 the least positive integer /such that 2/ = l m o d m 0 is 0(w0)so the ideal generated by 2 in Cmo is prime. Then this is the only prime ramified in Cm/C mo

and by Lemma 8 we get that 2)(hm for m = 24, 40, or 4 8. Since A f 4 = /zJ 0 =AJ 8 = l we havein fact that 2 J f h + so hm = h+ = l for these m .

33*



256 Masley and Montgomery, Cyclotom ic fields withuniquefactorizat ion

For m = 60 and 84 we consider C + / C + 0. For p\m0 the ramification index of p forC+JQ (also for C + / Q ) is equal to that of> for CmJQ. This occurs because m 0 =/?# where ;?and # are distinct primes and Cmo = C pC + 0 = C qC„ 0 so at most /? and at most q areramified for C m JC* 0 (the infinite primes, of course, ramify here). Hence, only primedivisors of the ideal generated by 2 in C+0 are ramified for C ^ / C ^ 0 . B ut for ra0 = 15, 21

the last positive integer/such that 2f= ±1 m o d r a0 is —~-.So the ideal generated by

2 in C+ 0 is prime . Since this is the only ramified prime for C + / C + 0 , L e m m a 8 applies äsbefore to give A60 = hS4 = A + 0 = /z 8

+4 = l.

In th e Situation that we have dealt with, we found that h+ = l whenever / z * = l.However, it should be noted that if K= Q(|/^T, ]/ ) then AJ = l, but h£ = 2 ([8], p. 123).Thus th e fact that the relative class number is one cannot then force th e real factor to beone, nor does it determine the parity of the real factor.

Bibliography

[1] N. C. Ankeny and S. Chowla, The class number of the cyclotomic field, Canad. Journ. of Mat h. 3 (1951),486—494.

[2] N. C. Ankeny, S. Chowla and H. H as s e , On the class number of the maximal real subfield of a cyclotomicfield, J. reine angew. Math. 217 (1965), 217—220.

[3 ] H . Bau er , N u m e r i s c h e B e s t i m m u n g von Klassenzahlen ree l ler zykl i scher Z a h l k ö r p e r , J . of N u m b e rTheoryl(1969), 161 — 1 6 2 .

[4] R. Brauer , On the zeta functions of algebraic number fields (II),Amer. J. Math. 72 (1950), 739—746.[5] H. Davenpor t , Multiplicative nu mb er theory, Chicago 1967.

[6] T. Estermann, Introduction to Modern Prime Number Theory , Cambridge 1961.[7] H. H as s e , Arithmetische Bestimmung von Grundeinhei t und Klassenzahl in zyklischen, kubischen und

biquadratischen Zahlkörpern, Abh. Deutsche Akad. Wiss. Berlin 1948 Nr. 2 (1950), 9 — 9 5 .[8] H. H as s e , Über die Klassenzahl abelscher Zahlkörper, Berlin 1952.[9] K. I w as aw a, A note on class numbers of algebraic number fields, Abh. Mat h. Sem. Univ. Hamburg 20

(1956), 257—258.[10] H. W. Leopoldt , Über Fermatquotienten von Kreiseinheiten und Klassenzahlformeln modp, Rend. Circ.

Math. Palermo (2) 9 (l960), 1 — 12 .[l 1] J. Mas ley , On the class number of cyclotomic fields, Diss. Princeton 1972.[12] T. Metsänkylä, On the growth of the first factor of the cyclotomic class number, Ann. Univ. Turku.,

Ser. A1155 (l972).

[13] //.L. Mo n tgo m ery and R. C. Vaughan, The large s ieve , M a th em a t ik a 20(1973), 119—134.[14] M. Newman, A table of the first factor for prime cyclotomic fields, Mat h. Co mp. 24 (1970), 215—219.[15] G . Schrulka v. Rechtenstamm, Tabel le d e r (relat iv-) Klassenzahlen von K r e i s k ö r p e r n , A b h . D e u t s c he A k a d .

Wiss. Berlin 1964 Math. Nat. Kl. Nr. 2.[16] C. L. Siegel , Zu zwei Bemerkungen Kum mers , Nachr. Akad. Wiss. Göttingen (1964), Nr. 6.[17] T. Tat u zaw a, On a theorem of Siegel, Japan. J. of Math. 21(1951), 163—178.[18] K. Uchida, Class numbers of imaginary abelian number fields. III, Tohoku Math. J. 23 (1971), 573—580.[19] A. Yokoyama, On class numbers of fmite algebraic number fields, Tohoku Math. J. 17 (1965), 349—357.

Department of Mathematics,University

ofIllinois

atChicago Circle,

Box4348,

Chicago,Illinois 60680,USA,

Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48104, USA

Eingegangen 15. Mai 1974

ciclotomic fields with unic factorizationrll.1976.286-287

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