Mathematisch Instituut Eoetersstraat 15
Amsterdam-C. The lietherlands
EUCLID'S ALGORITHM III CYCLOTOMTC FIELDS
II. W. Lenstra Jr.
Report γίφ-01
Received January 1J, 197U
EUCLID'S ALGORITHM TU CYCLOTOMIC FIELDS
H,W. Lonsbra, Jr.
Äbstract.
Eleven füll cyclotomic rings are proved to euclidean f'or the norn map.
AMS(MOS) subject classifi cation scheine (1970): 12A359 13F10S 1OE?0.
1
-Introductlon.
For a positive integer m , let ζ denobe a primitive m-tb root in
of unity. By φ we mean the Fuler (J-function. In this note we provc tho following theorem:
Theorem. Let o (m) < 10, m ^ 163 m ^ 2^. Then 22 [ζ Ί is euclidean for thc usual norm map.
Since 2Z [ζ J = 27, Γζ 0 for m edd5 this gives eleven non-isomorphic cuclidean rings, corr^sponding to
m = 1, 3, it, 5, 7, 3, 9S 1 1 , 1?, 15, 20. Thc cases m = 1, 3S i), 5, 89 12 are more or less classical Γ3, pp. 117-118 and pp„ 3£ 1-393; 10: 79
pp. 228-231; 53 chapters 12, 1U and 15; 6·, 9Ί . The ofcher five cases are apparently new=
For m even5 thj ring K Γ ζ 1 hao class number one if and only if Φ(Η>) s 20 or rn = 70} Qk or 903 see Γ8 l . So there are exactly thirty non-isomorphic rings TZ Γ ζ l vrbich adnit unique factori^abion. If sonie ^eneralized ^iemann hypotheses would hold, then all these thirty rings would be euclidean for some function3 possibly different from tho norm map Γ 1 1 1 .
Our notations are mostly Standard. For y , t . F and c see
in m m m
scction 1. By an overhead bar we denote the automorphism of φ(ζ ) which sends ζ tc ζ"" „ Gince the Galois group of φ(ζ ) over φ is abelian, liarring commutcs with all automorphi SIPS , tracrs and ncrms which we shall consider. This trivial remark will be constantly used without further mention. If we vicw φ(ζ ) äs a subfield of 0! 3 then barring is just complex con-jugation. The end (or absencc) of a proof is marked by Π.
The Gauss measure.
Let m ^ 1 bc an integer , and let t :ΐ(ζ ) ·* Q denote thc trace funetion t (χ) ~Σ σ(χ), the s um ranging over all automorphi sms σ of
m σ _
Ί(ζ ). The Gauss noasurc \^ :<\(ζ ) -> Q i s defined by l-^U) = ΐ^(χχ)9 cf, Γ3, p. 395; 1 I .
(1.1).(a) The function μ is a positive definite quadratic form onm the Q>-veccorspace φ(ζ ).
(b) For every real number r there are only finitoly many elemcuts y e Z Z [ r v l for whJch y (y) i r.
(c) For every χ e (β(ζ ) there i s < i y c TZ Γ ζ l such tho t μ (x+y) < μ (x+z) for all z e 71 U | .
~ 2
-Proof.(a) is evident frora μ (y) = ΐ σ(χ)σ(χ)5 and (b) follows from (a) m σ
sincc 2Z ι ζ j is α labtice in Φ(Γ )·m jm
Finally, (c) follovs froin (b) sincc /μ~~ saLisfies thc triangle inequality. D
Let th<= fundamental domain F b° de fincd byT,1
F^ = {x , aKrJ ! μη(χ-^) ^ ym(x) for ,.11 y c TL f ζη 1 }. ILen (l.l)(c) can b- rcsbatcd äs: F + TZ ' c,^ l ~ ^r^ '
Λ real nuirber c is called a bound for Γ if y (Ο ί c for al] x c F . It is•-'j^'—"~"-"~"-— Γ") ΤΩ. ΓΠ. easily secn t hat such c clo cxist. Clearly, th<. rc is a stallest bound foi" F }
which is denoted by c Tt is not hard to provo that v (x) ~ c for some in n m
x e F „ so that c is rational lut we rhall not nccd this . Λ bound c for m n s
F is called usai le if for evcry χ e l· sati&fyir" μ (χ) = c ther_ is
m m m
a root of unity u e 2Z ίζ l such that μ (x+u) ~ c, The use of usable bounds will becomc clear in tbe noxt section. Note that tvery c > c is a usable
m bound., since no χ e F satisfies μ (χ) = c > c .
a m m m
52. The euclidean algorithm.
Let Π = F : (ft( ζ ) -> C| be the norm function F (χ) = Π σ(χ) , the product ranging over the φ(ηι) automorphisms σ of φ(ζ ). For x e TL ίζ^ϋΧίΟ} we have
ΐ MX) l = |2Z ' r l/ /2 L ζ I x ! . We call Z L r; ' euclidean for thenorn if for 1 ' ' 'm τι ' m
every a, b c 7l [ζ l 5 b ^ 0 5 there are q, r c 2Z [r^ l such that a ~ qb+r and JK(r ) | < |N(b)| .
,1 . . .
Writinfc x - ab and y = -q, vo find., uning thc- multiplicativity of the norm:
(2,1). Tho ring ?Z l ζ Ί is euclidean for blu norm if and only if for ^very -i L· b ^m
x e Φ(ζ ) thcrf is an clcment y f T L i t , Ί such bhat |n(x+y)( < 1. Π m m
(2.2). For x e (ί(ζ ), ^π haven
Th<- oquality sig^ holas if and only if xx c (1). i—, f) ^
Proof. |IN(X)| - W(x)c = N(x)N(x) = F(xx) ~ Π σ(χ)σ(χ).
If we view φ(ζ ) äs a subfield of d s then o(x)r(x) is a nonnegative
real number for all o. Using th< arithmetic-geometric mean inequality we find ι 1 r. _/·._\_c_-\ \(Min) _ / 1 ,, /„\\c'(m)
σ χ σ χ
— 3
-Remark. Froip (2.2) one easily deduces: for χ ε ϋ Ζ Γ ζ Ί , χ ^ Ο , one has
U (x) ^ φ(η), the equality sig,n holdine if an«1 only if χ is a root of unity.
j_2·3). Lenma. Lot ->. e ι$(ζ ) be such that xx = (x+u)(x+u)= 1 for sone root of unitj u c ZZ Γζ ^ . Then χ c /? F ζ ί .
Frooj?. Put y = xu, then yy = 1 and y + y ~ 1 j S o y isa prirntive third root of unity. Then y e ^(ζ ) implies that m is divisiblt by 3= so
y e E Γ ζ 1 and χ = yu c 7L L ζ ' . 1n J m
1,2Λ). If φ(η) is a usable bound for Γ , bhen /Z Γζ Ί is euclidean for the norm.
Froof. Let χ e 3}(ζ,) "be arbitrary. lif havc to find an element y c ΕΓζ π such that |l\T(x+y)| < 1 . By (l.l)(c) we rnay essumo χ e T . Then
U (x) £ <,k(m), since φ (m) is a bound for F .
If the inequality is Etrict_ then |N(X) < 1 oy (2.2)9 and we can take y - 0. If the equality sign holds, then y (x) = ym(x+u) - φ(n) for somc root of unity u e 2Z Γζ \ since φ(η) is usable.
Then
If at least one inequality holds strictly, then vo can take y = 0 or y = u. If both equality signs hcld, then xx and (x+u)(x+u) are rationals by (2.2). Mor^over, L(XX) - 1 9 so we have xx ~ 1, and pimilarly (x+u)(x+u) = 1.
U sing (2.3) we find χ c K i ζ Ί , which contradicts χ e Fm since χ ^ 0. G
JJ3_V Estimating the fundaiaenbal domain.
JJ3» 1 ). Let n be a positive divi^or of n. Then c cm „ n__
- ^ .
Λ(ιιιΓ φ(τΓ f-l
J /· \ c.
Moreover if c is a usable bound for F a tUen m ··-· c is a usable Λ(ηΓ
bound for ί< ,m
h
-ί .3 · £_). · Leb χ e φ(ζ ) and y c- φ(ζ ). -Tuen HI n
J3. 3) o For χ e φ(ζ ) we have
Proof of (3.1), assuming (3.2) and (3.3).
2 Let χ e F be arbitrary. We have to prove μ (χ) i d . c . Applying (3.2) to y e Z Γ ζ ] and looking at the definition of F we find ~-b(x) e F . Sincc also χ. ζ3 € F for all j e S 3 we have
a n m m
in the. same vay — t(x. ζ""1) e F . Thercforc
for all j e ^ A ano1 using (3.3·) i t follows t hat μ (χ) < d^.c . m n This proves the first part of (3.1).
Next assumc c is a usable bound for F , and let χ £ F satisfy 2 n m
y Ix) = d .c, Then from the above reo-soning we see c = c and 1 ϊ
μ (— ΐίχ.ζ1*)) = c = c for all .] e ZZ . Take j = 0. n α m n
Since c is a usable bound for F , there is a, root of unity u e K Γ ζ "1 such that μ (— t(χ) + u) = c. Applying (3.2) with y = u we get
2 °
μα(χ+\ι) = μ (χ) - d .c« which prcvs that d'".c is a usable bound for F . Π
Proof of (3-2).
= d.tr(^- b(x)y + ^ t(x)y + yy) = tn(t(x)y) + tn(t(x)y) -f- d.tn(yy)
= tn(L(yy")) + tn(t(xy)) + tn(t(yy))
Proof Of (3.3). Let G be the Galoisgroup of <φ(ζ ) over φ(ς ).
In the computation bclov Σ and Σ refer to sumraations ovor G. We have σ τ
Lot ζ denote the m-th root of unity σ(ζ )τ(ζ )" . Then ζ = 1 σ>τ m m σ 5τ if and only if σ = τs and
lm „ ζ J = 0 if ζ * 1 j=1 'σ,τ ^σ,τ
·= m if C =· 1. σ,τ Itence the above expression becomes
t (Σ σ(χ)σ(χ) m) - m.t (t(xx)) ~ m.t (xx) = n.μ (χ) which proves (3.3)o ü
IJj. „Proof of the .thonroin.
Expiicit considorntior of the casc n = 1 shows that c, = r- is a usable bound for FI . Then ^ φ(η)^ i s a unable bound for F^, by (3 = 1). If
(^•1) φ(η) < lj 1 r)
then p φ(ιη) < φ(η), and φ (m) is a usable bound for F . By (2. M it follows that the ring ΖδΓζ^] is euclidean for the norm if (U,1) holds . This gives us exactly the cases m = 1 5 3, ^ ·, 53 3, 12 wbich mre already knovn. To get new cases ve use the folloving n?cults whJch will be proved in the next section,
l· Let n be a primf nurober. Then c = ··· -; - and thi s is a usable bound for F .
T,
suppose that m has a prime divipor r such that
Then a usable bound for F is ßivcn. by m
Φ (m;
c =
2? ι ζ "" i-i .uclirean frr tho norm, bj (".' )„
For which m, n does (k, 3} hold? In any case n|ra implies Φ (n) > φ(η) = n-1 so n+1 £ 1Γ Is nccessary. For n - 2 vc get o (n) -" } , which is (1<-.1). For n = 3 we have φ (τη.) < 6 which givos us thc new case m = 9 °
Für n - 5 wo fj_nri ^m) <- β which 10 Gatisfied ly m =· 15 arid ri - ?0. For n = 7 and n - 1 1 , finally, n - n satisfies (^.3). This provos thc thcorom, up tc (i|.2)„
±_Γΐ{ tcrmination of tho bound in a special cast.
Let n be an integer > 29 and Ict V be an (n-1 ) -dimensional
-vectorspace with gcncrators e- , 1 < i i n, suSject only to the relation =1 LT ~ Ot So for x· y- c K (1 < i i n) wo havo Σ^_ x.e. = Σ^_ y.e.^- I X l ~ ~ I J - 3 - - i - ~ " I X _ L
and only if χ. - y. -χ. - y. for all i , j . -^- *· U J
defjrie a positive dcfinitc quaclraLic form 0 on V by Q(x) - Σ . ( χ . - X ) 2 3 χ = Σ x.c. e V .
( 5 ): \Γ χ V -> ]R denote thc synmctric bi linear forn induced by Q: +y) Q(x)
-We have
(x,x) ~ Q(x) for χ e V 3 (e. , e. ) = n - 1 for 1 <" i < n, (e. , e.) - -1 for 1 < i <- j < n.
·*· J
L<=-t L r v be the subg^oup generated by {o. 1 < ί <" n} . Clearly, L is a lattice in V. Ifc put
F = {x r V | Qfx) < Q(x-y) for a] l y e l,} = fx c V | (x,y) < pQ(^ ) for τΐΐ y c L},
Since F i s compact, Q auswies a maximum c <_ u V s Ue are going to prove:
Thc set of Teints y c F for which Q(x) ~ c is given by {— Σ. ic / . χ Ι σ is τ, perriutation of {1,2, .... n}}
n i=1 σ(3 j l Moreovcr,
1? '
provc (Jj. 1 ) öfter a series of lentmas. A.t the end of this section we show (5.1) u
_ γ _
We call A proper if 0 5* A ϊ { 1 , 2, . . . , n} .
_( 5.3). Lemma. Let y e L be such that there is no A c { 1 5 2S . .. , n} with y - e.. Then bhere is an elcment z ~ + c. r L such tLat
A — J Q(z) -f- Q(y-z) < Q(y).
Proof. Let y - Σ. , ra.e. with m. e 7L . Using Σ / , e. « 0 we may assume — J L-"1 3 l l 1 = 1 l
0 < Γ._1 m. i n-1. For z = + e. we have 3(Q(y) - Q(z) - Q(y-z)) = (y,z) - (z,z)
= + (nm. - Σ1? , m. ) - (n-1). — j ι—ι ι
If "chis is >0 for some j and come choice of tho sign we are done. So suppose i t is <0 for all j and for both signs. Then for 1 < j < n we have
nm. £ Σ3? , m- + (n-1) < 2n-2 < 2n j i=1 i
nm. > Σ^_ m. - (n-1) > -n+1 > -n_ - " l
so m. i s 0 or 1 for all j, contradi cting bhat y has not the form e..Cl J A
(5. U). Let χ c V. Then χ e F <=> (x,c.) i sQ(eft) for all A c { 1 , 2, . . . , n} .A. Ά
Proof0 ·=-> is clear. <= : we Know
(x, eA) < gQ(eA) for all A c {l, 23 ,.., n} and we have to prove
(x,y) < ]Q(y) for all y e L,
T'his is done by an obvious induction on Q(y), using (5.3). Q
(5.5). Let x e F },e such that Qi'x ) = c. Then bhere are n-1 different — o o
proper cubseLs A(i) c f1, o, ..., n} , 1 i i ^ n-1, such that xo is the unique Rolution of vhe System of linear equalities
Proof„ Let
S = (Λ <= { I j 2, . . . , n) (XQ, e^) ~ ;Q(e^)} a
then we havr (x , e ) < ?Q(c. ) for all Λ c- {1, 2, . . . 3 η} , Λ <f S, o A -/\
8
-Λ ΐ ) is proper since e / . N ^ 0 . «Λ ι /
Therefore assume that fche linear span of {e. | A e S} has codimension A
in V . We derive a contradiction. The subspace
{z e V (z,e ) = 0 for all A e S}
has dimension at least 1 5 so for SOITK. z c V, ?, / 0 wo have (z, o ) = 0 for all A e S.
Replacing z by -z 5 if necessary, we may nssuric (5. T) (xo, z) > 0.
, multiplyinn z by a sufficiently small positive real nuniber we may assume
(z, eA) < aQ(eA) - (XQ, e^) for all Λ c f 1 s 2 , . . . , η> , Λ ^ S. Then for all A c { 1 3 2, . . . , n} we have
(xo + Z, eA) < JQ(eA)
so XQ + z e F by (S.M, But using (5. T) we find Q(XQ + z) > Q(xQ) + Q(z) > Q(xo)
contradicts our assumptiori Q(x ) ~ c = max {Q(x) | χ e F} . D
iöi. Let χ . A(1), o o „ 9 A(n-l) be äs in (5 = 5). Then A(i) c A(j) or A(j) c A(i), for all i, j , 1 < i < j < n-1 „
Fix i and j, anrt put A = A(i) and B = A(j). Let C = A\B and D - B\A. If G = 0 or D = 0 ve are done . So suppose C J- 0 Φ D . Then C n D
is equivalerit to
eAUB
so for X = A n B or for X = A u B ve have (x3e ) > 2Q(e ), contradictinp C J> Λ
}roof of (5.l)o Let χ e F satisfy Q,UQ) = c , and let A( 1 ),..., A(n-l) >e äs above. Froia (5.5) and (5.8) we conclude t hat {A (i) | 1 < i < n-1}
s a System of n-1 proper subsets of {1, 23 . ..s n} which is linear ly >rdered by incluQion. T'his is only possible if after a suitable renumberine <f the vectors e and the sets Mi) we have
A(i) = {1, 23 .,.., i} for 1 < i < n- 1 . ;y (5.5), we have
^e1 U0, e.) = JQ(eA(i)), for 1 < i < p-1 .
friting χ = Σ1} , x.e. in such a manner t hat Ση , χ. = Ο , we find 0 J=1 3 J J=1 J
xoa e.) = nx . . Also Q(c.) = A j . (n- |A| ) so our System becomcs tj J A
r., x . - O . learly, this implies
nx. = i(n+1 ) - i, 1 < i < n5 x = — Σ? , ie L. . .
o n 1=1 η+1-ι
re renumbered the e. at one point in the argianent , so it follows that XQ s in the set (5-2). Since there is at least one XQ £ Γ for which Q(XQ) = c, t follows by reasons of symmetry that conversely every element χ of (5=2) atisfies χ e F and Q(x) = c. Finally3 we
. . 2 n2-1
proves (5.1 ). D
of (k. 2). Lct n be a prime numbcr. The Q-vectorspacc ^(ζ) is _ κ
1=1 ζ = °- For rational numbers x., 1 < i < n, wc have
enerated by the n element s ζ\ 1 & i ^ n, subjcct only to the relation n
η
11 this implies that V can be considered äs TR ® „(ij^ Π J- Γ1 ΐ(ζ ), by e. = 1 ® ς ^ , D<i that Q is the natural extension of μη to V.
10 -2
Let n be prime. Then c = "' 9 and the s et of elements x e F n 12 n for which μ (x) = c is givcr ly
ί~ Σ· ι i ζ | σ is a perreutation of {1,2, ..., n}}„ G
To prove (i|.2)3 wo need only check usability of c . So iet x e F l ( ' \ "^ "^ atisfy μ (χ) = c „ Then x = - ü" i ζ ° for some permutation
n n n i-1 n of (1, 2, ..., n} . Hence
χ α(η) = 1 n-1 . σ(ά) = l ,r a(i-D n n 1=0 n n i=1 n
here σ(ο) = σ(η). By (5-9) it follows that μη(χ - ?n°^) = V proves that c is usable. Π
completes the proof of the theorem.
. The result (5.1) can also be descrihed äs follows. Let T = ffi /2Z >e a circle with circumference 1 , and for t ^ , ΐ0 c T Iet d(t1, tg) be ^e length of the shortest arc betveen t, and t„. For x = (x.)"_n e Tn Iet
l ίί. l X™ l q.(x) = min {Σ^=1 d(xia t)2 ) t e T} .
'hen
2 max (q(x) | x e Tn }= S_rl 9
-he maximutn being attained at those n-tuples of points "•j e T which divide T into n equal parts.
l'his follows from (5.1) and the identity
n Ση . r2 - E^. ., (r.-r.)? + (Ση_. r.)2, 1=1 L 1<i<j<n i j i=1 i -°ϊ· real numbers r. , .. . , r .
^ > * n
Let n be a positive divisor of m such that every prime which divides α also divides n. Put d = -= rfH · Then {1, ζ , ..., ζ^ά"1> is a
_ n Φ(η> m m
, l r ' is for TZ. { f, l and a straightforward computation (e.g„ using (3.3)) shows
V Σ?=ο ^ Φ β ά « ίο V*i>> f°r xie^n), 3 (3.l6)j„ All this implies c = d c a i.e.:r τη η J
2
wc \nou c0 ·= -r ai α c - - -., - for - J p irr-, H ΓΓ η r = fch-rt
c t = 22 , for t e ZZ , t s 1 ,
c, =—.2 ~ . p ~ .(p -1), for t , U £ Z 5 > 1 , p > 3 prime. 2 p 3
r
In particular c^ = 16 > φ(ΐ6) = 8 and c^ = 10j > φ (2*0 - 8, so our method does not apply to the cascs 16 and 2^4.
I do not know the cxact value of c if n has more than one odd prime m i
But using differcnt methods I can prove the follouing pertial converse to our theorem:
IJLuLl· Suppose φ (m) > 10 or m e {l6.,2k} . Then c > φ (m). D FJ
Of courses (6.3) docs not imply that the only valucs of m for which 2ΖΓζ ] is euclidean for the norm are giver by the theorem. In fact , I know of no principal ideal domain Z Γζ l which is prcved to be not euclidean for
m the norm.
(6·^). The ring 7L\t, f- ζ ~ l is euclidean for the norm ΠΠ . We show how this can be provod by our methods . Note that an element x = Σ._ χ. ζτ £ φ(ζ,1)3 with χ. e Q for 1 < i < 11 s belongs to
-L~" l l t l I I l
^(ζ^ + ζ ~ ) if and only if x. = x,.,_· for 1 < i s: 10.
Let x c (&(ζ + ζ ~ ) be arbitrary. ¥e have to oxhibit an element y e χ + ΚΓζ^ t- ζ~^] for which |N'(y)| < 1, where N> : ^(ζ1Ί + ζη ) -> φ is the norm.
From our proof that 2? ί ζ 1 is euclidean it follovs that there is an element y e P^, y e x -t- 27 Γζ^"1, such that |w(y)| < 1 ; here
N = Nn : φ(ζ ) -^ ($ is the norm. Write y ~ T^^ J^^ with y-^e φ for 1 < i < n. From y c x + 7L f ζ , , j w<_ deducc y. - y . e Z , forΊ l "L l l ™Ί
10. Also , ^ (y( ζ^ - ζ » l * TT by § 3
Hence y = y . for 1 ί i < 10, so y c Q U n + ζΤ). This implies y - x c J- l i ™~ l -< ι ί
^?n + ζ.,'1) η 2Ζ Γζ^Ι = ^ Γ ζ η + ς^], and S we find that y satisfies our requiroments .
An immediate gcneralization of this argunent yiclds :
if n < 11 is prime, then any integrally clcsed subring of 2Z l ζ l is — 1 ^
12
-<v.vovor, no nev renults are obtained in this vay: the cas° of quadritic rings is classical [59 eh. H'] , and more precise Information on the cubic rings ffi Γζ^ + ζ l and 2. [ζ + ζ ~ 3 can be found in [2] . I don't know whether my method applies to the ring Z [ζ + ζ ~ J , which was proved bo be euclidean for the norm in ΓΜ . Kote that the integrally closed subrings
ZZTg + l /_151 c 2Z [ζ J and ZS Γ/^Ü"] c ΣΖΓζ^Ί are not euclidean, since they are not even principal ideal domains .
(6o5). Throughout this note ve have used that complex conjugation comnutes with the other automorphisms of (|(ζ ). This is not essential for cur method. If K is any finite field oxtension of Q5 then the Gauss measure
Ug · K -> B > may be defined by y„(x) = Σ |σ(χ)|^3 the SUEI ranging over "the [K:Q] field homomorphisms σ: K -»· G . Then the main results of §§ 1-3 carry over to the general case.
Some care is required in stating (l.l)(a), since μ may assume non-rational values. The fundamental domain Fr, and the smallest bound CTA for F„ are
K K. Λ at F , ·. = F and c , ·, - c . But not be rational 3 and there is not necessarily an element χ € F for
in the obvious way, so that F , ·. = F and c , ·, - c . But c
μ (χ) - cv. Writing ETA for the ring of algebraic integers in K, we Λ. K K
can generalize (2. h) äs f ollovs .
If CK : Q] is a usable bound for FTr, then P is euclidean for the
rv K
norm. Π
generalization of (3.1) reads:
0 Let K be a finite field extension of Q, and let m e 22 , m > 1.
Lot L be a field extension of the form L = K(a), where α = a e R . 2
Suppose there is a real nuniber r > 0 such that a(a) = r for all field homomorphisms σ : K -> (Ϊ. Then
2
0 CT < — c,, Σ1?"1 r1^, L ία Κ 3 =o
where d = PL ; Kl . Morcover, if c is a usable bound for F 3 then ehe right hand siue of (6.8) is a usable bound for F . Finally, if ία 0 < i < m-1} is an R -basis for R, (so that in particular
K i-j then the equality sign holds in (6.8).
- 13
-The validity of (6 „6) and (6.7) is not affected if the concept of a "usable bound" (end § 1 ) is veakened äs f ollows :
a bound c for F is usable if for every χ e P for which μ (χ) = c there exists a unit u e R„ such that yr,(x+u) = c.. - g _K
°nly the proof of (2,3) nceds a small modificabion .
'· J.W. S. CasselSj On a conjecture of R. M, Robinson about sums of roots of unity, J. Reine Angov.Math. 238_( 1969 ) 1 12-131 .
• H. Davenport 3 On the product of three non-homogerteous linear forms9 Proc . Cambridge Philos.Soc. h3_ (19^7)137-152.
3- C.P. Gauss, Werke,, Zweiter Band, Göttingen 1876.
4« H „ J „ Godwin, On Euclid's algorithm in some q.uartic and quirtic fields, J„ London Math. So c. ho( 1905)699-704 .
5, G.H. Hardy and E.M. Wright, An intr,oductiqn to the theory of numbers, Oxford 1938 , 19^5 , 1951» , I960 .
"» R.B. Lakein, Euclid's algorithm in complex quartic fields , Acta Arith. 20 (1972) 393-^00.
'· E. Landau, Vorlesungen über Zahlentheorie, Band 3, Leipzig 1927.
ö· J.M. Masley, On the dass number of cyclotomic fields, thesis, Princeton University 1972.
9· J.M. Masley, On cyclotomic fields Euclidean for the norm map, lotices Amer.Math.Soc. _1£ (1972) p.A~8l3 (abstract 700-A3).
V - J. Ouspensky, Note sur les nombres entiers dependant d'une racine cinquieme de l'unite, Math.Ann.66(1909)109-112.
'· P.J. Weinberger5 On Euclidean rings of algebraic integers5 Proc.Symp. Pure Math, gi-i, Analytic Number Theory., Amer. Math. Soc. 1973,
