97
Internat. J. Math. & Math. Scl.
(1986) 97-104
Vol. 9 No.
RELATIVE INTEGRAL BASIS FOR ALGEBRAIC NUMBER FIELDS
MOHMOOD HAGHIGHI
Department of Computer Science
Bradley University
Peoria, Illinois 61625 U.S.Ao
(Received January 23, 1985 and in revised for March 26, 1985)
ABSTRAC’I’.
for O
OK
a
K
()-1
a
At first, conditi)ns :re given for existence of a relative integral basis
()[-i
with [K;k]
OK4(ai{’ 2 )/()k(Z;3)
f,>r
Finally
ex.[stence and construction of the relative
()K()(n,/-3) l()k3(n),’)k6(n,
basis
KEY
Tlc:n the constrtiction of the ideal I in
n.
is given for proof of existence of a relative integral basis for
8
lOk2()
/Z)
integral
for some values of.n are given.
1,’zt,:t,.l!/
W(.)iI)S ANi) PIIRASI,S.
discri,,nant, c.Zaas
1980 A?.L’.;
1.
SL/BJE(,’i’ CLASS.IFICA?’/ON (7ODE.
P,imary 12A30, 12F05
1NTi/OI)UCF I()N
Throughout this article the following notation will be used:
Q:
field of rational numbers
Z:
rational integers
k,K (QC kCK): algebraic
nu,,ler fields
disc(x):
d.iscriminant of element x
DK/k:
relative different: of extension K/k
Oki
h
ritg o[ intt:ger’s f k.
0
class number for k
k
P.I.
principal
NK/ka:
relative nor, of an ideal a in K for extension
2,,
FINITEI,Y GENEITED
n
[l, p. 24] i
was sim that f H is a finitely generated module over a
Dedekind ring R then
M
R
m
I
A
(2.1)
is an Ideal. in R, A is a torsion-sdmodul.e and m
where
Now for extension K/k witi IK;k]
n, by (2.1)
s
-l.
so by tli.;
wtr
(2 2)
Iave:
’I"ltI{()IEM 2.3.
over O
a positive integer.
we have
In tle extesion K/k for
P.I.
IKlkl
n, 0
K
has relative integral basis
M. HAGHIGHI
98
2.4.
II,I,IJS’I’I,’A’iI()N
of
l,et
()(/2,’-7)/.j
-)(/-!,)
SOI,U’I’IN,
y (2,,2),
Q(2),
k
xi(?
k
Q([’-7).
2
Does a relative integral basis
121.
;1;,
"eitivc i.tegral basis" exists
P.I., otlmrwise
IlOt.
We will construct
(2-, -1)= 1,
tll,i
,n
ideal
it,
I.
z
,,c
y__].
(dKl,dK2)
Since
-.
[1"
,-
wheFe R :is an
O
1-. *
O
where
K
usillg u Cltcorem given in [3, p. 21.8],
.’&
%
(’k 3
,R,
Ok-mOdule
Rs [i-g
prctportional; tlen
We take (2.5) and (2.6)
1
/-’+
u
2t,
2;
(2.)
+ f++-4
(2.6)
+ 2tCUl =-7 + C7 + u/’]- +-14-v
-8
8
-
+u+
2
+ -14v
for
v
u
0
t
s
-4
Th erl
/
4 + 0
]- ()+"7)
[2,2J1
72
12,/ZiI
Since
over
I)].
’l’ll,
is not P.l. in
!,)/i1
ideal
[2,JL]I
)
O]
is an ideal in 0
k
does not have
relative integral
K
(li) to eqivalmce of idea.Is).
titan 0
is liuiqlie
’l’le merited ul tlu l.)reviols theorem is only good for th case n
n
1)
co,q))/.;)tion )f an i(l;l
in 0
O
k
[1,]]l-z
-1 )
2 since for
is too difficult. hus e need
and one of the invariants in the extension
relati()n sucl aS tlt) following I:)t?tw(’t.tl
K/k.
TIIEORI’I 2.7.,
If C is the class of ideal.s in k containing
dK/k
and
CK/k
is a
c].ass contaili.g l, then
C
C..
K k
Now we will give the "criterion for existence of a relative integral basis", for
the extension K/k.
]’HEORtI 2,,8.
over O
k
<=>
[1, p. 359].
dK/k
See Norki.ewicz ct al.
Let [K:kl
n, h
[,4,5,6].
odd, then O
has a "relative integral basis"
k
K
(relat:ive discri.ninant) is a principnl ideal. For more details see
RELATIVE INTEGRAL BASIS FOR ALGEBRAIC NUMBER FIELDS
PROOF:
2.7
Theore,
If
dE k
is P.I.
P 1.,
dK/k
"-:=:
(2,1 k)
since
OK/O k
=>:
llas a relative integral basis,
very ideal in the class of
s,
1. ’l’ien 1= P.I. and by (2.2), O
P.I.
99
Therefore by
K
12=
is P.I. Therefore
(’K/k
has a relative ntegral
P.I.,
bsis
over 0
k
COROL1.ARY 2.9.
Theorem 2.8
P.I.I)., a
If O
for every
I,et k
Does a rel.atiw basis of
0
, 2b
.,ab
3
(a,b)
for
-.z
1.,
(-3,
2 ).
c;ses
b
.2
b
a
,:1 e
(-3,
311
See
5,
2i3
-6)
32
f()
ab.
1
in k
Q(),
h3
21.
3
and
for
-z
a
fo
so
312-313 31 311-31. 2
1)6/3= 311., d6/3= 311 31 d6/3 31
Iy Tleorem 2.5,
221].
K6
and Type II..
n k2
3
so 3
3 and e define
3ab
}11
31
.2
lelim,
Q(,),
Q(),
rood 9
rt.,:pecLively Type
3
311 }1
2 ,2 2
3].3.12313
3
In Type II, 3
31"32
ab
/a b
call iese wo
3
and k
exist: or not
()0
In Type 1,
d6/3
Q(2])
3
OK 6 ]1""3
kno that for n=al, 2
()3
e
dK/k
ntegr! basis exists.
re]ati.ve
II.LUSTiATION 2.10.
e
is odd and
P.l..l)., tlen h k
P.I. Thus by
of k where the ring of integers is its
k
lignite extension
(-343,2]-6)
snce
(3-6)
s
h
a
3
is odd and
P.I., so a relative ntegral
bas i. exists.
lci.lntlly in (3.1)
integral1 basis over
we wi.l] prove that [f 3
lt lere I
03,
21 so
3
31h 3
then O has a reltiue
6
3
and also a relative integral
h
basks exists.
EXISTENCI,: )F A RELATNIVI INTI’,GRAI, BASIS:
3.
BY SOME COND[TIONS ON n FOR
O6(,723)/ O3().
Now ieru we ill stow iar for
sonde
n
z
this extension has relative integral
ba s s.
TiIE()RI.,;M
I,
If 3
3.1.
ttn
Type 1.
PROOF.
O
3
BuE
and
I)6/3/ Z 311/
(-3, l) whe,,
.2
(3)
(31.". 2) lr ’l’yl,’
]
h
(-3,
31
In Type [[, (3)
cae
3
.
so
is a P,1. in O
(-3,
31
and
3
for
6 generated by an element of k 3.
when
in Type I
3lab
2.
(-3,n)
or
2
(31-32),
1)
31
3
(31)3
(3) or
is P.].
Then
it is dependenE on deaLs
(-3,)3
1/31
31
(3) for P.I.,
also generates a P.I. in 0
and
32
therefore
6
$n this
a relative bais may ex.it, or ,ay not exlst.
But i.t may be har
in k
ab
(31)3 (-3,
l)
has relative i.ntegra] basis over O
2221, 0 6 hs a relative integral basis over
1/31
3
3
Now, 3
so:
,
By Tleorem in [l,
OK6
3
exits.
((2i-3),
i,
3
’l’lerure tle
’,n
,n
,
3l!
21 ,,,I
nd
l,t
0
(123-
’l
ivc, rs
Int
aain
has relative basis over 0
6
6),
so
3l
2.[
and
,
3. or example
relative integral basis
! Tiere 3.] is nor tre in generl.
k
3
):
M. HAGHIGHI
I00
Tlll,]Oll".bl
3
h
3
3.2.
of Lilt: iollwing statements holds:
!)
n
3
2)
n
p
n
3p or 9p,
4)
n
5)
n
5 (rood 9)
and q
p-q (p,q are pr:imes), p
2
p .q (p,q are distinct primes), p=-q :-2 or 5 (rood 9).
3)
<=
I)F:FtNI’I’IIN
)ne
p
.
p
prime
-1 (rood 3)
(rood 9)
2 or 5
prime
2
n,uber n :is called a tlonda number if i.t i.s a number in the
A
3.3.
p
table for Theorem 3.2
By Theorems 3.l and 32 we have:
’I’tlEOIEH 3.4.
n is a
re].ative integral basis over
4.
REI.ATIVE
INTEGRAl,
llonda
O6(, )
number in type I,
o3(n).
OK6(,i-3) / OK3 ().
BASIS OF
We proved in Theorem 3.1 that if 3
Therefore a relative integral basis for
h
3
06/0
then
is P.I. only for Type I.
31.
since by the theorem
3 exists,
3+
3+’
[3, p. 2011,
integral, bass fr
5.
=31
disc 1,
06
over 0
31
dK6/k 3
Therefore
is a relative
31
3
CONSTaUC’rON O .,A’rv INTE(;aL aSS FOR
Since
]ntu;
O.2(,=1
and if
311
06(n,)/02()
is P.I.I., tl,en by (2.3)
b;l; ’.
TIIIORI",bl
For Type I and
l,et
(1,
31
is a P I.
if
3]a
then
tl’n
PR)OF.
If
relative
has
3
5.l.
02
)6
x
necessarily has a
Since
d6/2
3,.{,2/ A,
N6/2(3/ab2/,)
a.d
3 ’6-
N6/2(3/a2b/ 2)
are in
02
9
then are integers
3/ab2/,, Jab/X")’02 then by the theorem
3/.,2b/ .’1’ is ; relative integral basis of 06/0.,,
disc(l,
to compt,te disc(x).
d is{: x
2
34
2
6
31. 31
,2
31 3’p p2
o
in [3, p. 201]
so
we are going
-
I01
RELATIVE INTEGRAL BASIS FOR ALGEBRAIC NUMBER FIELDS
i’
(ab)"’(3+,-3)
6
{ 31 (302-3"P)+31
2
3.+.32 )262
For "’l’yl)e I" w.c.
il;vt,
()6/(),}.
integral l)tsis ()1
5.2.
ll,l,tlS’rlCA’rtt)N
(Type I,
31a),
tlisc x,
;c’t
(:()illl et
[!,
x
s()
17,9,10].
1.
(}(2i3),
K
Vc)r
--’
mc,tio, t:l,t
il
A;s,tme
3’ItI:tRI.:M 5.3.
_J
,2
31
is t
k
t?(
3
O
where
2
(31P 2_73 .10)+3’1’2(31.0 -310) j2
2
3--2-I)/A’]
ll)2/A,/;i
(2i
the i(leal 3
so
6
We iave to
2
l()
rio/3
,2
is a re.lutive
-6)
=
!.1., this is still an ot,n question.
1,2),
(3)
(1
31
n)
for
ab and
"Typ [",
,--
t6
-1 for u
1.
PROIIF.
;,tEe
for a= b= 3k+l. and
I)ec;)use
"]
}+,b 2 + } ,2b
31
31
NK6/K2((
NK6/K2((t2)
;nd
2
.+
t
ntegc’rs. e take x
are
<7+
2
+
0
disc(x)
p
2
i]/ul) 2 431
3
vza b"
)ab 2
p
2
3
3
t]
t
2
2 /-
l’(t2ab+ t2
+ 31"(02ul.,+t3
+
3
3-2A, b+
+ p2
’ab +
2
3
/ah-
4-
’:)I) 2
4-
2.
31 (i>2ab+ pt 3
disc(x)
))
’a"b +
0
].
2
9
3-2-
/a b +
2
3
3-
2
:
)
/2 b + t 1
/a-b +
2
t2r, /a21)+
3
+,
31.
+
3
31
{],a].,,:2
31
3
for
(5.4)
2
Now
4-tl-
2
k
3k+2.
b
JllLc’r;.ll
ald
_]
3k+2 or convursely and
3k4-1 at,l b
0 f,r a
where
0
31
2 3-<,b’+
2
p rib- p
"t -p ab
3
3
2
2
t
3
:)
/ul,-+t2,, 2 }i.,+-i)+
2
ab-t
"t-O
2
3
3
b-
t3P
t
2
ab 2- 2 ")a2b t 2 3)
/a2----Ot 3 )ab2+t2"t3
t
3)
M. HAGHIGHI
102
nab(o’-P)-3bl:2(O-,. O) 3ab(t)2-P)(n-t2)=
di.sc(x)
disc(x)
1/3
disc(x)
.2
’l’l,s
2
3
2
a2b 2
’! 2
9
(5.4) is
f"()
K 6 /k
(!) If k
)(
OK6
over
n
2
),
(+ 1)
3
then n
3k+2
and a
0
5-
a
ogo
’I’Iit,;(i!,’I.:M
31
a
is ccsarily P.I. so for such n e can
’l’lic relal, iv int..r,r;l basis .in "Tyix., If" of
%-6.
00?
lb2
b
rood)
+
0
00
()ai., I)i/’3 i,., an i,,teg,,,l, be,’,,,,se:
(ab2) 3= (]- t+ 1) 3, so -3/2 (t3-t)
(b2-1)2 2 1+2ab 2 0
6
+ 4 -t27
[l,t,Ool,
1;
+
For ’"l"ype I1" (i.e. a
0
consgruc[
a
(5.4) for O K /K
h 2
OK 6
I’IIIOF.
k2
31.
hd numbt.rb
relative integral basis as
x
.
()k2.
O.
(1
eqatin
2
-.-
(3ab)-= ,lis(x)
result [-r the case
s;me
so
Fur al
’
’)
,.I
,.;in,’u
IIA,IS’I’RA’I"ION 5.5.
3k+l,
2
rt’ltiv’ ite,,r;l b;,is ir
Also we lave tte
b
3
3ab(p -o)(n- n+
9)
2
2b
3
+
O_._
Fro,,,
ab
(’),,b 2
2- l+9t
2
hich shows
so
over O
Ok2
+3
1+
0
OK6
en it
)//25
K
2
sat sfes
n
is an ntegral and also
t, e
and at last e have the equation:
s
an ntegral.
e
take
tlcn
disc(x)
O0
-0)
2
a2b 2
-}J?-3yx
)2
II,I,USTRATION 5.7.
lh,r
O16
k
3
2
2
Since
+ 1-
/ab
3,..-.-2
/K 3
fo
ab2+ l)+ O0
(ab)2
(0
disc(x), then
is a relative integral basis of
Q(),
:’-
a
b rood 9, so
3
ab 2- 1-)ab2+])-] 2
Ok2
OK6/Ok
2
RELATIVE INTEGRAL BASIS FOR ALGEBRAIC NUMBER FIELDS
gJve anotllc:r
Itere we wil
over
OK2
fr
TIIi.X)REH
27
3t+l
n
5.8.
tie,>rem 17or corrq)uting a relative integral basis of
3t
!,
is a P.I. in O
k
be square-free in k
-n
m
or
31
no matter wlletimr
l.et n
103
OK6
not.
Q(n) for Type I,
3
then
oF
(j_,)3
3
3t
)/.-5 (3t
3
+ ,C!)
3t(l-n)
’)
9
(l-n
2
-n
-3nt
n
(l-n)
3
+
3
3/---3
0
9t2(1-n
or bri.el ly:
27(1.
)2
6
wlich shows that
6
2/t
4
2
4
9t (l-n)
(1-.) (2n+.l)
+ (’2n+l) 4 +
(’l-n)2
2
(l-n)4
(l-n)
6
0
0
is an integra]. Now we take
[
+
Z::
+
/n2
n
2-/n
d s (: (x)
(02_ 0
)
2
R I’.’F E R I’:N CE S
1.
NAR[EWICZ, W.
1976.
2.
B[I,’I), R.II. and PAltRY, C.,I. lntegral Bases for Bicyclic Biquadratic Fi.elds over
Qadratic Sd)field, !l!...!:J._!!.-.(1976), 29-36.
3.
l5Nl, P.
4.
Elementary and Analytic Theor olgebraic N,mbers, Pwin, Warsaw,
.
.Jj!L!jf.21p).)f_’..)yfLD,_,
3ohn Wi].ey & Sons, New York, 1972.
AR’I’IN, E. Q,estions dv I$,se Fliima.l dans ]a Tleori.e
Nombres Algebriques,
Nation;2_J... .i. J’2_._l...gfl’FJ.?...5f-_’..._[.[j.I,k}., XXIV (1950), 1.9-20.
M. HAGHIGHI
104
5.
Ste.initz (’l,s: of (yelic Extensions ,,f Prime Degree, J. Reine
I,()NG, R.I,.
A.,SY.. ..’.- ..9
1921 ), 87-98.
(;rl; ].t.)t:ltlX llrmt I"ris, ]963.
6,,
SEII’;E,
7.
COLIN, li.
8.
itONDA, T.
9.
IAIRUCANI, P. and CtlllN, tl. Remarks on Principal Factors in a Relative Cubic
:i,l,I, ..1:. 2’P?-..’’[’gP.F#’ ! (1971), 226-239.
.J.[’.
A
;i;i:l
Ivit:tio t.o Al;t,lraic Numbt, rs and Class Feld
’l’heor.,
Ihtre Cbic Fields wh;e Class Numbers are Htl. tiples of Three,
Se,’,,,i
}.?".Y?..i.., t!J’:’t.’’sP.r],
John Wiley, New York, 1962.