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computing the Galois group
Ines Abdeljaouad, Annick Valibouze
To cite this version:
Ines Abdeljaouad, Annick Valibouze. The Hacque method and the complete GI-method for computing
the Galois group. [Research Report] lip6.2000.025, LIP6. 2000. �hal-02548331�
InesAbdeljaouadandAnnikValibouze y
CalFor-LIP6,UniversitParisVI,
4PlaeJussieu,F-75252PARISCEDEX05
E-mail: abdeljaomediis.polytehnique.fr;avbmediis.polytehnique.fr
1.INTRODUCTION
TwophilosophiesareproposedforomputingtheGaloisgroupofaunivari-
ate polynomial: thealgebraioneand the numerialone. Thenumerial
methods provide eÆient algorithmsevenwith ahigh degreeof preision
(see [19℄, [7℄, [13℄); the algebrai methods guarantee an exat result in
reasonable time (see [21℄, [6℄) and the work undertaken in [3℄ and [4℄ is
deterministi.
Weproposetwoalgebraimethods for omputingthe Galois groupofan
irreduiblepolynomialf whihwealltheHaquemethod(in[12℄)andthe
ompleteGI-method. WestartbyintroduingtheHaquemethod: wegive
in setion 2asystemof harateristiequationsof the Galoisgroup asa
subgroupofthelinearalgebraigroup. TheompleteGI-method isintro-
duedinsetion3. ItomputestheGaloisgroupthankstoanalgorithmof
omputationofthedeompositiongroupoftheidealofrelations(see[20℄).
Throughout thisartilek indiates aeld of harateristizeroand f an
univariatepolynomialirreduibleoverk. TheHaquemethodforomput-
ingtheGaloisgroupoff annotbeusedwithoutpreliminaryomputation:
infat,aminimalpolynomialofaprimitiveelementoftheGaloisextension
must be omputed. Thus, setion 4is devoted to transform the Haque
method into an implementable form. For this, we ombine the Haque
methodwiththerststepsoftheompleteGI-methodandwenallyom-
parethistwomethods.
*
NotesInformellesdeCalulFormelnumero2000-08,presentationoraleaAAECC'13
enNovembre1999
y
SupportedbytheprojetGaloisoftheUMSMEDICIS,Palaiseau,Frane.
2. GALOIS GROUP ASA SUBGROUP OFGL
N (K)
Let K be a nite extension eld over k of degree n > 1. Let us x
e = (e
0
;e
1
;:::;e
m
), a basis of the k-vetor spae K, suh that e
0
= 1
andm=n 1.
2.1. Notationsand Denitions
Theringofk-endomorphismsoverK isdenoted byL
k
(K)and thegroup
ofthe invertibleelementsofL
k
(K)byGL
k
(K). TheGalois group ofthe
extensionkjK is,bydenition,thegroupofk-automorphismsoverK. It
isdenotedbyGal
k (K).
Weset M
n
(k)tobetheringofnn matrieswithoeÆientsin k. We
denote by M[:;e℄ the isomorphism of algebra whih assoiates with any
k-endomorphismofL
k
(K)itsmatrixin thebasise:
M[:;e℄ : L
k
(K) ! M
n (k)
f 7! M[f;e℄ .
We denoted by GL
n
(k) the group of the invertible matries of M
n (k).
Then
GL
n
(k)'GL
k
(K) : (1)
For all 2 K, we denote by b
the multipliative endomorphism of
over K dened by b
(x) = x, for all x in K. Remark that the eld
b
K=f b
2L
k
(K)j2Kgisisomorphito K.
LetK=fM[
b
;e℄2M
n
(k)j2Kg. TheeldK is naturallyisomorphi
totheeld b
K. Thus,weobtainthefollowingisomorphisms:
K' b
K'K : (2)
Thegroup of theinvertibleelements of K (resp. of b
K)is labeled byK
(resp.
b
K
). The group b
K
is isomorphi to K
and, it is a subset of
GL
k (K):
b
K
=f b
2GL (K)j2K
g : (3)
We set K
= fM[
b
;e℄ 2 M
n
(k)j 2K
g. Then K
GL
n
(k) and we
havethefollowingisomorphisms:
K
' b
K
'K
: (4)
LetGandH betwogroupssuhthatH G. ThenormalizerofH inG,
denotedbyNor[G;H℄, isequalto:
Nor[G;H℄=fa2GjaHa 1
=Hg :
Definition 2.1. Letg2GL
k
(K). Theappliation g isalled K-semi-
linear ifforallx 2K and2K,there exists ans2Gal
k
(K)suh that
g(x)=g(x)s().
2.2. Propertiesof the Galoisgroup Gal
k
(K)as a subgroup of
GL
n (k)
Proposition 2.1. The Galois group Gal
k
(K) is the set of the K-semi-
linearappliations g of GL
k
(K)suhthatg(1)=1.
Gal
k
(K)=fg2GL
k
(K) j gis K semi linear andg(1)=1g : (5)
Proof. Wenote that forall g2Gal
k
(K),g isK-semi-linearand g(1)=
1. Conversely, let g be a K-semi-linear appliation suh that g(1) = 1,
we have to show that 8x 2 k, s(x) = x. For any x 2 k, there exists
s2Gal
k
(K)suhthatg(x)=s(x):g(1)=s(x)=x. So,g2Gal
k (K).
The following proposition is a onsequene of lemma 2.1 of [11℄, and we
givehereadiret proof:
Proposition 2.2. The normalizer of b
K
in GL
k
(K) is equal to a set
ontaining allthe K-semi-linearappliations ofGL
k (K).
Nor[GL
k (K);
b
K
℄=fg2GL
k
(K) j gisK semi linearg : (6)
Proof. Let g 2 Nor[GL
k (K);
b
K
℄ be a k-endomorphism, then for all
2 K
, gÆ b
Æg 1
2 b
K
. For all2 K
, there exist 2K
verifying
gÆ b
=bÆg ; letsbeanappliation from K
toK
suhthat s() =
; s is abijetion ofK
beause it isa surjetionof K
. Toprovethat g
isK-semi-linear,weanonlyprovethatg veriesg(x)=g(x)s()where
s2Gal (K).
(i) For 2 K
, we have g Æ b
= d
s()Æg ; then for any x 2 K,
gÆ b
(x)= d
s()Æg(x)andthusgÆ b
(x)=g(x)=g(x)s().
(ii) Letus verifythat the bijetions is ak-morphism ofK. Weset ,
2Kandx2Kthen,aordingto(i),g(x( + ))=g(x)s( + ). Inaddi-
tion,g(x(+))=g(x)+g(x)=g(x)s()+g(x)s()=g(x)(s()+s()).
Thus,likeg6=0,s(+)=s()+s().
In the same way, g(x()) = g(x)s() and g(x()) = g(x)s() =
g(x)s()s(). Sine g 6= 0, we obtain s() = s()s(). Lastly, g(1) =
g(1)s(1)andthus s(1)=1.
So,s2Gal
k
(K)andtheappliationg isK-semi-linear.
Conversely, let g be a K-semi-linear appliation. Let prove that for all
2K
,gÆ b
Æg 1
2 b
K
.
Bydenition, for 2K and x 2 K,there exists s 2 Gal
k
(K) suh that
g(x)=g(x)s(). Partiularly,foreah2K
andx2K,gÆ b
Æg 1
(x)=
gÆ b
(g 1
(x))=g(g 1
(x))=g(g 1
(x))s()=xs(). SogÆ b
Æg 1
= d
s() .
Sine2K
ands2Gal
k
(K), theelements()is invertible(i.e. s()2
K
)andwededuethat theappliationgÆ b
Æg 1
2
K
.
Theorem2.1. Aording tothe propositions 2.1and2.2, wehave:
Gal
k
(K)=fg2Nor[GL
k (K);
b
K
℄jg(1)=1g : (7)
Thankstothe isomorphism (1),the Galoisgroupasasubgroup ofGL
n (k)
isexpressedinthe following form:
Gal
k
(K)=fA2Nor[GL
n (k) ;K
℄jA(e
0 )=e
0
g: (8)
2.3. Charaterization ofGal
k
(K)with a systemof equations
WeseekasystemofequationswhihharaterizestheGaloisgroupGL
n (k).
For that,in allthispart,wexAamatrixofGL
n (k) .
Lemma2.1. If the matrixAof GL
n
(k) veriesA(e
0 )=e
0
thenwewrite
itinthe form:
A= 0
B
B
B
1
1;0
:::
m;0
0
1;1
:::
m;1
.
.
. .
.
. .
.
. .
.
.
0 ::: 1
C
C
C
A
(9)
where
i;j
2k for (i;j)2[1;m℄[0;m℄ anddet(
i;j )
i;j2[1;m℄
6=0 .
Proof. Obvious.
2.3.1. Charaterization of the eldK
Forj 2[0;m℄, letset M
j
=M[eb
j
;e℄the matrixof eb
j
in thebasise(eb
j is
the multipliative endomorphism of e
j
). Let 2 K and 0
;:::; m
2 k
suh that= P
m
j=0
j
e
j .
Thewriting
M[
b
;e℄= X
m
j=0
j
M
j
(10)
givesaharaterizationoftheelementsoftheeldK . Furthermore,be-
longsto theeldK
ifandonlyif j
(j2[0;m℄)arenotallzero.
Thus,(M
0
;M
1
;:::;M
m
)isabasisofK
.
2.3.2. Charaterization of Nor[GL
n (k) ;K
℄
The matrix A belongs to Nor[GL
n (k);K
℄ if it veries AK
A 1
= K
.
Thisisequivalentto:
8i2[1;m℄ ;9(
i;0
;:::;
i;m )2k
n
f(0;:::;0)g AM
i
= X
m
j=0
i;j M
j A:
(11)
Corollary2.1. LetA2GL
n
(k)and(12)thelinearsystemof equations
dedued from(11):
AM
i
= X
m
j=0 x
i;j M
j
A; i2[1;m℄ ; (12)
where x
i;j
are unknown. The matrix A belongs to Nor[GL
n (k) ;K
℄ if
and only if the system of equations (12) admits atleast one solution =
(
i;j )
i2[1;m℄;j2[0;m℄
ink mn
.
2.3.3. Charaterization of the Galoisgroup
Theorem2.2. ThematrixAofGL
n
(k)belongstotheGaloisgroupGal
k (K)
(a) itiswritteninthe form(9),
(b) thesystem(12)admits atleastonesolution.
Proof. Aordingtothetheorem2.1,thelemma2.1andthesetion2.3.
Definition 2.2. Let B = (b
i;j )
i;j2[1;n℄
be a matrix of M
n
(k) where
b
i;j
are unknown. Let us put X = (x
i;j )
i2[1;m℄;j2[0;m℄
where x
i;j
are also
unknownentries. Thefollowingsystemofequations:
B(e
0 )=e
0
and BM
i
= X
m
j=0 x
i;j M
j
B ; i2[1;m℄ (13)
isalled thesystemofequationsof the Galois group Gal
k
(K)in the basis
e.
Corollary2.2. AmatrixA belongstothe Galois group Gal
k
(K)ifand
onlyif the system(13)of equationof Galois group inthe basise admitsa
solutionB =A andX = for aertain2k mn
.
2.4. Simpliation ofthe systemof equations ofGal
k (K)-
Haque system
Letu2K beaprimitiveelementof theextensionk jK (i.e. k(u)=K)
andletu n
(a
m u
m
+:::+a
1 u+a
0
)beitsminimalpolynomialoverk.
For j2[0;m℄, weanput e
j
=u j
. WedenotebyM
0
thematrixidentity.
Inthebasis(1;u;:::;u m
),thematrixM
1
oftheendomorphismubiswritten
by:
M
1
= 0
B
B
B
B
B
0 0 ::: 0 a
0
1 0 ::: 0 a
1
0 1 ::: 0 a
2
.
.
. .
.
. .
.
. .
.
. .
.
.
0 0 ::: 1 a
m 1
C
C
C
C
C
A
Andforj2[0;m℄,weset M
j
=M
1 j
.
Lemma2.2. The system (13) of equations of Galois group in the basis
(1;u;:::;u m
)isequivalentto:
B(e
0 )=e
0
and BM
1
= X
m
x
j M
j
B ; (14)
wherex
j
2k for allj2[0;m℄ .
Proof. Itisobviousthatthesystem(13)involvesthesystem(14). Reip-
roally,leti2[2;m℄andsuppose that(14)isveried. Then
BM
i B
1
=(BM
1 B
1
) i
=( X
m
j=0 x
j M
j )
i
= X
m
j=0 y
j M
j
;
where y
j
belongs tok[x
0
;:::;x
m
℄ beause(M
0
;M
1
;:::;M
m
) isabasisof
K
.
Lemma2.3. If a matrix A 2 GL
n
(k) and = (
0
;:::;
m
) veries the
system(14), (i.e. B =A and(x
0
;:::;x
m
) = are solutions), then A is
writtenin theform(9) and
j
=
1;j
for j2[0;m℄.
Proof. Tobe onvined, it is enoughto express AM
1
and M j
1
A for j 2
[0;m℄, inthebasis(1;u;:::;u m
).
Theorem2.3 (Haque). AmatrixAbelongstotheGaloisgroupGal
k (K)
ifandonly ifthe matrixA isinvertibleand
A= 0
B
B
B
1
1;0
:::
m;0
0
1;1
:::
m;1
.
.
. .
.
. .
.
. .
.
.
0
1;m :::
m;m 1
C
C
C
A
suhthat AM
1
= X
m
j=0
1;j M
j A
where
i;j
2k for (i;j)2[1;m℄[0;m℄.
Definition2.3. Thesystemofthetheorem2.3isalledHaquesystem.
2.5. Exampleof Haquesystem
LetF =T 6
+108andA2GL
6
(Q)suhthatA(e
0 )=e
0
,where(e
0
;:::;e
n )
is theanonial basis of GL
6
(Q). The Haque systemfor the Galois ex-
tensionoverQofthepolynomialF(T)=T 6
+108isequalto:
AM = M A+ M A+ M A+ M A+ M A+ M A ;
where:
M1= 2
6
6
6
6
6
6
4
0 0 0 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 108 3
7
7
7
7
7
7
5
and Mi=M i
1
for i2[0;5℄ :
ThustheHaquesystemis equivalenttothefollowingsystemof 30equa-
tionsand30unknowns:
3;0
1;0
2;0
=0,
4;0
1;0
3;0
=0,
5;0
1;0
4;0
=0
108
5;0
1;0
5;0
=0,
2;1 2
1;0
1;1
=0,
2;0
2
1;0
=0
3;1
1;0
2;1
1;1
2;0
=0,
4;1
1;0
3;1
1;1
3;0
=0
5;1
1;0
4;1
1;1
4;0
=0, 108
5;1
1;0
5;1
1;1
5;0
=0
2;2 2
1;0
2;1
2
1;1
=0,
2;3
1;0
2;2
1;1
2;1
2;1
2;0
=0
2;4
1;0
2;3
1;1
3;1
2;1
3;0
=0,
2;5
1;0
2;4
1;1
4;1
2;1
4;0
=0
1082;5 1;02;5 1;15;1 2;15;0=0,
3;2 21;03;1 21;12;1=0
3;3 1;03;2 1;12;2 2
2;1
3;12;0=0,
3;4 1;03;3 1;12;3 2;13;1 3;13;0=0
3;5 1;03;4 1;12;4 2;14;1 3;14;0=0,
4;2 21;04;1 21;13;1 2
2;1
=0
1083;5 1;03;5 1;12;5 2;15;1 3;15;0=0,
4;3 1;04;2 1;13;2 2;12;2 2;13;1 4;12;0=0
4;4
1;0
4;3
1;1
3;3
2;1
2;3
2
3;1
4;1
3;0
=0,
4;5
1;0
4;4
1;1
3;4
2;1
2;4
3;1
4;1
4;1
4;0
=0
108
4;5
1;0
4;5
1;1
3;5
2;1
2;5
3;1
5;1
4;1
5;0
=0:
5;2 1;05;1 1;1(4;1 1085;1) 2;1(3;1 1084;1+116645;1) 3;1(2;1
1083;1+116644;1 12597125;1) 4;1(1;1 1082;1+116643;1 12597124;1+
136048896
5;1 )
5;1 (
1;0 108
1;1
+11664
2;1
1259712
3;1
+136048896
4;1
14693280768
5;1 )=0
5;3
1;0
5;2
1;1 (
4;2 108
5;2 )
2;1 (
3;2 108
4;2
+11664
5;2 )
3;1 (
2;2
108
3;2
+11664
4;2
1259712
5;2 )
4;1 (
2;1 108
2;2
+11664
3;2
1259712
4;2 +
136048896
5;2 )
5;1 (
2;0 108
2;1
+11664
2;2
1259712
3;2
+136048896
4;2
14693280768
5;2 )=0
5;4 1;05;3 1;1(4;3 1085;3) 2;1(3;3 1084;3+116645;3) 3;1(2;3
1083;3+116644;3 12597125;3) 4;1(3;1 1082;3+116643;3 12597124;3+
1360488965;3) 5;1(3;0 1083;1+116642;3 12597123;3+1360488964;3
14693280768
5;3 )=0
5;5
1;0
5;4
1;1 (
4;4 108
5;4 )
2;1 (
3;4 108
4;4
+11664
5;4 )
3;1 (
2;4
108
3;4
+11664
4;4
1259712
5;4 )
4;1 (
4;1 108
2;4
+11664
3;4
1259712
4;4 +
136048896
5;4 )
5;1 (
4;0 108
4;1
+11664
2;4
1259712
3;4
+136048896
4;4
146932807685;4)=0
108
5;5
1;0
5;5
1;1 (
4;5 108
5;5
)
2;1 (
3;5 108
4;5
+11664
5;5 )
3;1(2;5 1083;5+116644;5 12597125;5) 4;1(5;1 1082;5+116643;5
1259712
4;5
+136048896
5;5
)
5;1 (
5;0 108
5;1
+11664
2;5
1259712
3;5 +
1360488964;5 146932807685;5)=0
TheGaloisgroupofF(T)=T 6
+108isthesetofmatriesAverifyingthe
Haque system. After theidentiation of this systemwithsubgroups of
GL
6
(Q),wededuethattheregularrepresentationoftheGaloisgroupof
F overQis isomorphitoS
3 .
We will see, in setion 4, that the identiation proess of Galois group
usingHaquesystemisaeleratedwhenweusetheresultoftherststeps
ofGI-method.
3.THECOMPLETE GI-METHOD
Letf be apolynomialoverk ofdegreen and
f
bean-tuple of nroots
off in analgebrailosure
^
k of k. We proposean algorithmwhih om-
putesthedeompositiongroupofagivenidealandweprovethat, applied
totheidealof
f
-relationsI
f
, thealgorithmomputestheGaloisgroup
Gal
k
(K)=G
f .
3.1. Notationsand denitions
We denoted by S
n
the symmetri group of degreen and I
n
the identity
group of S
n
. For 2 S
n
and = (
1
;:::;
n
) a n-tuple in
^
k, we put
:=(
(1)
;:::;
(n)
). Wedenotebyk[x
1
;:::;x
n
℄theringofpolynomial
in thevariablex
1
;:::;x
n
overtheeld kand k[T℄thering ofpolynomial
inthevariableT overtheeldk.
Theationofthepermutationgroupofdegreenonk[x
1
;:::;x
n
℄isdened
by:
S
n k[x
1
;:::;x
n
℄ ! k[x
1
;:::;x
n
℄
(;P) 7! :P(x
1
;:::;x
n
)=P(x
(1)
;:::;x
(n) ) :
For 2 S
n
, a n-tuple in
^
k and P 2 k[x
1
;:::;x
n
℄, wehave: :P() =
P(Æ). LetJ beasubsetofk[x
1
;:::;x
n
℄and2S
n
then(J)=f:P j
P 2Jg.
Definition3.1. LetLbeasubgroupofS
n
andH asubgroupofL. The
polynomial2k[x
1
;:::;x
n
℄ isanL-primitive H-invariant if
H =f2Lj:=g :
Definition3.2. Theidealofthe
f
-relations,denotedbyI
f
,isdened
by:
I
f
=fP 2k[x
1
;:::;x
n
℄ j P(
f
)=0g :
Definition3.3. TheGalois groupof
f
is denedby:
G
f
=f2S
n
j8P 2I
f
; :P(
f
)=0g :
3.2. Galois Ideal and deompositiongroup
Let=(
1
;:::;
n
)bean-tuplein
^
k. A polynomialP 2k[x
1
;:::;x
n
℄is
an-relation ifP()=0.
Definition3.4. LetLbeasubgroupofS
n
. TheidealI L
ofL-invariant
-relationsdened by:
I L
=fR2k[x
1
;:::;x
n
℄ j (82L):R ()=0g ;
isalledan(L;)-Galoisideal.
TheidealI S
n
isalled theideal of symmetri relations and, aordingto
denition3.2,I I
n
=I
,theideal of -relations.
Example 3.1. Letf bea polynomial overk of degreen and
f be a
n-tupleofnerootsoff inanalgebrailosureofk. Ife ;:::;e represents
thenelementarysymmetrifuntions, thenthepolynomialse
1 e
1 (
f ),
...,e
n e
n (
f
),alledCauhymodulusoff,formaGrobnerbasisofI S
n
f .
Definition3.5. ThedeompositiongroupGr(I)ofanideal Ik[x
1
;:::;x
n
℄
isdenedby:
Gr(I)=f2S
n
j(I)=Ig :
Remark3. 1. Gr(I)isagroupanditveriesthefollowingequality:
Gr(I)=f2S
n
j(I)Ig :
Remark 3. 2. Aording to thedenition 3.3, theGalois groupof
f is
thedeompositiongroupoftheidealI
f of
f
-relations:
G
f
=f2S
n j(I
f )=I
f
g=Gr(I
f ) :
Remark 3. 3. WehaveI G
f
f
=I
f
andifH is asubgroupof G
f then
I H
f
=I
f .
3.3. Computation of the Deompositiongroup ofan ideal
Theorem3.1. Let g
1
;:::;g
s
be generators of anideal I in k[x
1
;:::;x
n
℄.
The deomposition groupGr(I)of I isthe biggest subgroup Gof S
n veri-
fying:
8i2[1;s℄ and 8j2[1;r℄
j :g
i
2I ; (15)
where
1
;:::;
r
aregeneratorsofG.
Proof. Letg
1
;:::;g
s
beageneratingsystemoftheidealI ink[x
1
;:::;x
n
℄.
Let 2Gr(I) then (I)=I, in partiular foreah generatorg
i
of I we
have(g
i
)2I. Thus,Gr(I)veriestheondition(15).
2S
n
suhthat:
8i2[1;s℄:g
i
2I : (16)
Let g 2 I then, g
1
;:::;g
s
are generators of I so, there exist t
1
;:::;t
s in
k[x
1
;:::;x
n
℄suhthatg=t
1 g
1
+:::+t
s g
s
. Thus:gisalsoalinearombi-
nationof:g
1
;::::g
s
overk[x
1
;:::;x
n
℄. Thus,usingidentity(16),wehave
:g2I forallg2I. Then 2Gr(I).
Weproposean algorithmalled IDG whih omputesthe deomposition
groupof anidealI dened byaGrobnerbasisofI. Italsoneedsalistof
groupswhih ontainstheDeomposition groupof I. Thislistis alled a
CandidateListanditanontainsthesymmetrigroupS
n .
Thefuntion return(G)givesus thedeomposition groupGoftheideal
I and the funtion Add(G,g) adds thepermutation g to the set G. Let
g
1
;:::;g
s
beaGrobnerbasisoftheidealI andLasetofallthegenerators
ofthegroupsintheCandidate List.
Algorithm 1(IDG(<g
1
;:::;g
s
>,L)).
1. begin
2. forf 2I do
3. G:=fg
4. forg2L do
5. ifg:f2I then Add(G,g); end if
6. endfor
7. L:=G
8. endfor
9. return(G)
10. end.
ThegroupGisthedeompositiongroupGr(I) oftheidealI.
Proof. In eah step of the algorithm, the group L dereases. In fat,
this grouponvergestowardsthe deomposition groupof theideal I and
aording to theorem 3.1, the algorithm IDG swithes o in a nished
numberofsteps.
Thestep5. ofthealgorithmispossiblewhenwetakeaGrobnerbasisofI.
Infat,g:f2I ifandonlyiftheremainderof theredutionof g:f under
theidealI isequaltozero.
ofthealgorithman bedonebyremovingallthefatorsoftheelementsg
inL.
3.4. Determinationof thegenerators ofI
f
for the
omputationofG
f
The algorithm IDG applied to the ideal of
f
-relations gives the Ga-
lois groupof G
f
. So, in order to ompute the G
f
using the omplete
GI-method, we initially haveto omputea Grobnerbasis of the ideal of
f
-relations.
Arstmethod,duetoN.Yokoyama(see[16℄),onsistsongivenafatorof
thepolynomialf insomesuessiveextensionofkuntiltheeldofdeom-
position off. Thedisadvantageofthismethod, appliedtoourproblem,is
thatitsostisveryhigh.
TheGI-methodisloselyrelatedtotheomputationofG
f
(seeAlgorithm
4.2in[20℄). It onsistsonomputingthegeneratorsof Galoisidealsusing
relativeresolvents. Itisthemethodexposedinthis paragraphin orderto
omputetheidealofrelations.
Definition3.6. Let2k[x
1
;:::;x
d
n℄andLasubgroupofS
n
ontain-
ingG
f
. TheL-relative resolvent of
f
by istheunivariatepolynomial
overkgivenby:
L
;
f
;L
= Y
2L:
(T (
f )) :
WhenL=S
n
,thisresolvent,denotedbyL
;f
,isalledtheabsoluteresol-
ventof f by .
Definition3.7. LetH andL betwosubgroupsofS
n
suhthatH L
and let be anL-primitiveH-invariant. The invariant isL-separable
for
f
ifandonlyif(
f
)isasquare-freerootof theresolventL
;f;L .
WhenL=S
n
,is alledseparablefor
f .
LetEk[x
1
;:::;x
n
℄. TheidealgeneratedbyEink[x
1
;:::;x
n
℄isdenoted
by<E >.
Theorem3.2 (Valibouze). Let H and L be two subgroups of S
n suh
thatH LandG
f
L. LetbeanL-primitiveH-invariantL-separable
for
f
andlet F be a minimal polynomial of (
f
)over k. Then I H
f
=
I L
f
+<F()>.
By assumption, the polynomial F is asquare-free fator, irreduible over
k,of theresolvent L
;
f
;L .
Proof. Seetheorem3.27in[20℄.
Now,rststepsoftheGI-methodprodues(likefortheHaquemethod)a
groupLontainingtheGaloisgroupG
f
,italsogivesI L
f
andCandidate
ListalistofsubgroupsofLandidatetobetheGaloisgroup. Weompute
a polynomial F square-free fator, irreduible over k, of the L-resolvent
L
H
;
f
;L
,whereH andverifytheonditionsoftheorem3.2andH an
be anelementof CandidateList. So, weredue theCandidate List using
theorem3.2. IfmoreoverH isasubgroupofG
f
then, byremark3.3,we
have: I
f
=I L
f
+<F(
H
)>,itisinpartiulartheaseofH =I
n . Ifthe
CandidateListontainsoneelementthenitistheGaloisgroup,otherwise,
weomputeaGrobnerbasisforthelexiographiorderofI
f
usingavery
fastalgorithm developed byJ.C. Faugre (seeFGBin [8℄ or[9℄ for more
detailsonGrobnerbasis)andweapplythealgorithmIDGtothisGrobner
basisandto Candidate Listin orderto omputetheGaloisgroupG
f of
f.
4. THECONSTRUCTIVE HACQUE METHOD
Letf beasquare-freeunivariatepolynomialoverk ofdegreedand letK
be itsdeomposition eld overk. The eld K is ofdegreen assupposed
inthepreedingsetion.
We see in setion 2.4 that the Haque system, annot be implementable
without theminimal polynomialof a primitive element of the Galois ex-
tensionofk.
TheGaloisresolventallowsustoomputeaminimalpolynomialofaprim-
itive element ofk jK, but aswe will see in thissetion, its omputation
ispratiallyimpossible. Furthermore,tobeeetive,theHaquemethod
should not haveimpratiable preonditions. Thus, we searh to take a
partiular fator of the Galois resolvent to determine a minimal polyno-
mialofaprimitiveelementoftheGaloisextension.
4.1. Minimal polynomialof a primitiveelementof kjK
In order to ompute a primitive element of the extension eld k j K or,
moreexatly,itsminimal polynomialonk,thehistorialmethod onsists
polynomialf (see[21℄). Infat,anysquare-freefator,irreduible overk,
of this resolventis the minimal polynomialof aprimitive element of the
extensionkjK. Thisresolventbeingof degreed!,it isquiteobviousthat
itsomputationisdoomedtofailureoverthedegreed=6.
Theidea,presentedhere,istoomputeonlyonefatorin koftheGalois
resolvent in order to redue theomplexityof the problem. For this, we
willuserelativeresolvents denedbelow.
Definition4.1. LetV 2k[x
1
;:::;x
d
℄. AresolventL
V;f
isalledGalois
resolvent if
ithasonlysquare-freeroots,
V isanS
d
-primitiveI
d
-invariant.
Proposition 4.1. There always exist many polynomials V suhthat the
resolvent L
V;f
isaGalois resolvent. ForsuhaV,eahrootofthe Galois
resolvent isaprimitive element ofthe algebraiextensionkjK.
Proof. Sinek isaperfetinniteeldandf issquare-free,see[10℄.
Remark 4. 1. With the assumptions of denition 3.6 and for an L-
primitiveH-invariant(HL),theL-relativeresolventL
;f;L
isofdegree
[H:L℄anditis afatoroftheabsolute resolventL
;f .
LetusonsiderapolynomialV 2k[x
1
;:::;x
d
℄suh thatL
V;f
beaGalois
resolventandletF oneofitssquare-freefatorirreduibleoverk. Without
lossof informations, we ansuppose that V(
f
) is one of theroots ofF
andthus,F ishisminimalpolynomialoverk.
IfLis asubgroupof S
d
ontainingtheGaloisgroupG
f
thenbyremark
4.1 thedegreeof theresolventL
V;f;L
isthe order ofL and L
V;f;L is a
fatoroverkoftheGaloisresolventL
V;f .
IftheorderofthegroupLissuÆientlysmall,itispossibletoomputethe
resolventL
V;
f
;L
(see [15℄,[18℄ andtheGI-methodin [20℄). Thus, tond
suhgroupLitisneessarytoapplyrststepsoftheompleteGI-method
untiltheomputationoftheGaloisidealI L
f .
4.2. ComparingHaque methodand CompleteGI-method
Reall that the GI-method is the algorithm of [20℄ whih produes the
ideal of -relations and alist, alled Candidate List, ontaining groups
andidate to be theGalois group(see setion 3.4). Ifthe Candidate List
ontainsonlyoneelement,itistheGaloisgroupG
f .
As the Haque method requires therst steps of GI-method to ompute
somegroupL,itisnaturaltoomparetheHaquemethodandtheomplete
GI-method(see setion 3). Infat,supposethat rst stepsof GI-method
omputesaGaloisidealI L
f
andaCandidateListwhihontainsagroup
Lverifyingonditionsofsetion4.1.
Let set V anL-primitive I
n
-invariant. For the Haque method, werst
must omputeand fatorize the resolvent L
V;f;L
. Next, weidentify the
Haque system (see theorem 2.3) with some group of Candidate List to
havetheGaloisgroup.
Besides, the omplete GI-method (see setion 3) omputes the resolvent
L
V;
f
;L
. After that,wegiveaGrobnerbasis ofthe idealof
f
-relations:
I
f
=I L
f
+<F(V)> whereF, aminimal polynomial ofV(
f
)overk,
is a square-free fator, irreduible over k, of L
V;
f
;L
. But, we prefer to
omputeandfatorize theresolventL
H
;
f
;L
where
H
isanL-primitive
H-invariantandH asubgroupof L ontainedin G
f
(the groupH may
be found usingCandidate Listsee also theorem 3.2 andsetion 3.4). So,
here,weompute relativeresolventsof degreesmaller thanthe degreeof
theresolventL
V;
f
;L
omputedintheHaquemethod.
Furthermore, after theomputation of theidealof
f
-relations,wemust
omputeaGrobnerbasisofitinorderto applythealgorithm IDG.
5.EXAMPLE OFCOMPUTATION OFTHEGALOIS
GROUP FOR N =6
For n = 6, let f = x 6
+2. The GI-method applied to f gives a list of
groupsandidatetobeGaloisgroupoff (thislistontainssomesubgroups
ofPGL(2;5)),italsogivestheidealI
PGL(2;5)
f
(see[20℄):
I PGL(2;5)
f
= <24x6+x 3
3 x
3
2 x1+8x
3
3 x
2
2 x
2
1 +6x
3
3 x2x
3
1 +5x
3
3 x
4
1 +
8x 2
3 x
3
2 x
2
1 +4x
2
3 x
2
2 x
3
1 +8x
2
3 x
2 x
4
1 +6x
3 x
3
2 x
3
1 +
8x3x 2
2 x
4
1
4x3x2x 5
1
+12x3+5x 3
2 x
4
1
+12x2+14x1;
24x
5 5x
3
3 x
4
2 7x
3
3 x
3
2 x
1 16x
3
3 x
2
2 x
2
1 7x
3
3 x
2 x
3
1
5x 3
3 x
4
1 8x
2
3 x
4
2
x1 12x 2
3 x
3
2 x
2
1 12x
2
3 x
2
2 x
3
1
8x 2
3 x
2 x
4
1 12x
3 x
4
2 x
2
1 16x
3 x
3
2 x
3
1 12x
3 x
2
2 x
4
1
+8x
3 5x
4
2 x
3
1 5x
3
2 x
4
1 2x
2 2x
1
;
24x4+5x 3
x 4
+6x 3
x 3
x1+8x 3
x 2
x 2
+x 3
x2x 3
+8x 2
3 x
4
2 x1+4x
2
3 x
3
2 x
2
1 +8x
2
3 x
2
2 x
3
1 +
12x3x 4
2 x
2
1
+10x3x 3
2 x
3
1 +4x3x
2
2 x
4
1
+4x3x2x 5
1 +
4x
3 +5x
4
2 x
3
1 +14x
2 +12x
1
;
x 4
3 +x
3
3 x2+x
3
3 x1+x
2
3 x
2
2 +x
2
3
x2x1+x 2
3 x
2
1 +
x
3 x
3
2 +x
3 x
2
2 x
1 +x
3 x
2 x
2
1 +x
3 x
3
1 +x
4
2 +x
3
2 x
1 +
x 2
2 x
2
1 +x2x
3
1 +x
4
1
;
x 5
2 +x
4
2 x1+x
3
2 x
2
1 +x
2
2 x
3
1 +x2x
4
1 +x
5
1
;
x 6
1
+2> :
the Haquemethod
LetV =x
3 +2x
2 +3x
1
beaPGL(2;5)-primitiveI
6
-invariantomputed
with the algorithm PrimitiveInvariant in [1℄ (see [2℄ and [14℄ for the
omputation of primitiveinvariants). The relative resolventof f byV is
omputedwiththegeneralizationofalgorithmin [17℄(see[5℄):
L
V;
f
;PGL(2;5)
= (T 12
+15444T 6
+343064484)(T 12
21164T 6
+188183524)
(T 12
572T 6
+470596)(T 6
3456) 2
(T 6
+128) 2
(T 6
+2) 2
(T 12
+1012T 6
+19307236) 2
(T 6
54) 4
:
TheresolventL
V;
f
;PGL(2;5)
ofdegree120isafatoroftheGaloisresol-
ventL
V;f
ofdegree6!=720. Theomputationtimeand thefatorization
of theresolvent L
V;
f
;PGL(2;5)
is immediate (less thantwoseonds). Let
F =T 12
572T 6
+470596beasquare-freefator, irreduibleoverQ,of
L
V;
f
;PGL(2;5)
. So, theGaloisgroupisatransitivegroupoforder 12and
G
f
PGL(2;5).
TheHaquesystemof F is asystemof 12 2
=144equationsand asmuh
unknowns. By identiation in the list of andidate ontaining all sub-
groupsofPGL(2;5),theGaloisgroupofF andforf isisomorphitoD
6
thedihedralgroupofS
6 .
the CompleteGI-method
Itis suÆienttoomputeadisriminantresolventofdegree20insteadof
theresolventofdegree120forHaque. Infat,weomputethePGL(2;5)-
resolventassoiatedto
C6
aPGL(2;5)-relativeC
6
-invariant,whereC
6 is
a yli group of order 6 in S
6
. We also ompute a Grobner basis of
I
f
=I
PGL(2;5)
f
+<F(
C6
)>,whereF isasquare-freefator,irreduible
overQ,ofthePGL(2;5)-resolventassoiatedto
C .
6.CONCLUSION
TheHaquemethodisanewapproahoftheGaloistheoryandithara-
terizesit withasystemof equations. TobeeÆient, this method anbe
usedinthenalstepoftheompleteGI-method.
TheompleteGI-methodmirrorsthedesentmethodofStauduharandwe
an saythat the diereneis that thetest forrationalityof anevaluated
invariantisreplaedbyatest forinvarianeofanideal. Itisneessaryto
omputeaGrobnerbasisoftheidealofrelationstoobtaintheGaloisgroup.
In this ase, if the degree of the Galois group is reasonable, it would be
preferabletouseHaquemethod(theHaquesystemwillnotbesolarge).
Otherwise,liketheexampleof setion5,it issometimes moreeÆientto
omputeadisriminatingresolventthatwillreduethelistCandidateList
tooneelement: theGaloisgroup.
The omplete GI-Method is also used to ompute the ideal of relations
whih allows us to make algebrai omputations on the splitting eld of
thepolynomial. So,ifwewanttoomputeonthesplittingeld,theom-
pleteGI-methodisthebest method.
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