109 On the quartic residue symbols of certain fundamental units A. Azizi et al.
Frontiers in Science and Engineering - Vol. 6 - n° 1 - 2016
An international Journal Edited by The Hassan ii Academy of Science and Technology
On the quartic residue symbols of certain fundamental units
Abdelmalek Azizi 1,4, Mohammed TAOus 2 and Abdelkader zekhnini 3 1- ACSA Laboratory, Sciences Faculty, Mohammed First University, Oujda, Morocco.
2- Faculty of Sciences and Technology, Moulay Ismaïl University, Errachidia, Morocco.
3- Pluridisciplinary Faculty of Nador, Mohammed First University, Morocco.
4- Corresponding Author E-mail: abdelmalekazizi@yahoo.fr
A. Azizi et al. On the quartic residue symbols of certain fundamental units
On the quartic residue symbols of certain fundamental units
Abdelmalek Azizi1, Mohammed Taous2and Abdelkader Zekhnini3
1ACSA Laboratory, Sciences Faculty, Mohammed First University, Oujda, Morocco.
2Faculty of Sciences and Technology, Moulay Isma¨ıl University, Errachidia, Morocco.
3Pluridisciplinary Faculty of Nador, Mohammed First University, Morocco.
4Corresponding Author E-mail: abdelmalekazizi@yahoo.fr
Abstract. Letd be a square-free integer,k = Q(√
d, i) and i = √
−1. Letk(2)1 be the Hilbert 2-class field ofk,k(2)2 be the Hilbert2-class field ofk(2)1 andG= Gal(k(2)2 /k)be the Galois group ofk(2)2 /k. We give necessary and sufficient conditions to have Gmetacyclic in the case where d = pq, withpandq are primes such thatp ≡ 1 (mod 8)and q ≡ 7 (mod 8)and the2-class group ofkis of type(2,4), using the quartic residue symbols of certain fundamental units.
Key words:2-class groups, Hilbert class fields,2-metacyclic groups.
1. Introduction
Letkbe an algebraic number field andlbe a prime; thenlkwill denote a prime ideal ofkabovel.
We denote, also, by x
lk
the quadratic residue symbol for the primelkapplied tox. Ifkcontains i=√
−1andlkis prime to2, then we can define x
lk
4
by : x
lk
4
≡xN(lk4)−1 (modlk).
WhereN(lk)is the absolute norm oflk. LetCl2(k)denote the2-class group ofk,k2(1)its Hilbert 2-class field andk(2)2 its second Hilbert 2-class field that is the Hilbert 2-class field ofk(1)2 . PutG= Gal(k2(2)/k)andG its derived group, then it is well known thatG/G Cl2(k). An important problem in Number Theory is to characterize the structure ofGusing the residue symbols, since the knowledge ofG, its structure and its generators solve a lot of problems in number theory such
Frontiers in Science and Engineering
An International Journal Edited by Hassan II Academy of Science and Technology 1 A. Azizi et al. On the quartic residue symbols of certain fundamental units
On the quartic residue symbols of certain fundamental units
Abdelmalek Azizi1, Mohammed Taous2and Abdelkader Zekhnini3
1ACSA Laboratory, Sciences Faculty, Mohammed First University, Oujda, Morocco.
2Faculty of Sciences and Technology, Moulay Isma¨ıl University, Errachidia, Morocco.
3Pluridisciplinary Faculty of Nador, Mohammed First University, Morocco.
4Corresponding Author E-mail: abdelmalekazizi@yahoo.fr
Abstract. Let dbe a square-free integer,k = Q(√
d, i)and i = √
−1. Let k(2)1 be the Hilbert 2-class field ofk,k(2)2 be the Hilbert2-class field ofk(2)1 andG= Gal(k(2)2 /k)be the Galois group of k(2)2 /k. We give necessary and sufficient conditions to haveGmetacyclic in the case where d = pq, withpand q are primes such thatp ≡ 1 (mod 8)andq ≡ 7 (mod 8)and the2-class group ofkis of type(2,4), using the quartic residue symbols of certain fundamental units.
Key words:2-class groups, Hilbert class fields,2-metacyclic groups.
1. Introduction
Letkbe an algebraic number field andlbe a prime; thenlkwill denote a prime ideal ofkabovel.
We denote, also, by x
lk
the quadratic residue symbol for the primelkapplied tox. Ifkcontains i=√
−1andlkis prime to2, then we can define x
lk
4
by : x
lk
4
≡xN(lk4)−1 (modlk).
WhereN(lk)is the absolute norm oflk. LetCl2(k)denote the2-class group ofk,k(1)2 its Hilbert 2-class field andk2(2)its second Hilbert 2-class field that is the Hilbert 2-class field ofk2(1). PutG= Gal(k(2)2 /k)andG its derived group, then it is well known thatG/G Cl2(k). An important problem in Number Theory is to characterize the structure ofGusing the residue symbols, since the knowledge ofG, its structure and its generators solve a lot of problems in number theory such
Frontiers in Science and Engineering
An International Journal Edited by Hassan II Academy of Science and Technology 1 A. Azizi et al. On the quartic residue symbols of certain fundamental units
as capitulation problems, whether class field towers are finite or not and the structures of the 2- class groups of the unramified extensions of k withink(1)2 . In this paper, we give an example of this situation.
Letk=Q(√pq, i), wherep ≡1 (mod 4)andq ≡3 (mod 4)are two different primes such that
q p
= 1, then the symbols εq
p
:=
εq
pQ(√q)
and
ε2q
p
:=
ε2q
pQ(√2q)
do not depend on the choice ofpQ(√q) andpQ(√2q) (εm is the fundamental unit ofQ(√
m), wherem =q or2q).
According to [1],2εm is a square inQ(√
m)wheneverm= qor2q, so εm
p
= 2
p
. Ifp≡1 (mod 8), then we can define the following quartic residue symbol:
εm p
4
:=
εm pQ(√m,i)
4
(see.
[8]). In the present paper, we give explicit expressions of this symbol form = q and we show that if the2-class group ofkis of type(2,4), then it takes the value−1and we also show that the metacyclicity ofGis characterized by the value of this symbol form= 2q≡14 (mod 16).
Letmbe a square-free integer andk be a number field. Throughout this paper, we adopt the following notations:
• h(m), (resp.h(k)): the2-class number ofQ(√
m), (resp.k).
• εm: the fundamental unit ofQ(√m), ifm >0.
• Ek: the unit group ofOk.
• Wk: the group of roots of unity contained ink.
• ωk: the order ofWk.
• i=√
−1.
• k+: the maximal real subfield ofk, ifkis a CM-field.
• Qk= [Ek:WkEk+]is the Hasse unit index, ifkis a CM-field.
• Cl2(k): the 2-class group ofk.
2. The quartic residue symbol of ε
mand applications
In what follows, we adopt the following notations: if p ≡1 (mod 8)is a prime, then
2 p
4 will denote the rational biquadratic symbol which is equal to1or−1, according as2p−14 ≡1or −1 (mod p). Moreover the symbol p
2
4 is equal to (−1)p−81. A 2-groupH is said to be of type (2n1,2n1, ...,2ns)if it is isomorphic toZ/2n1×Z/2n2×...Z/2ns, whereni ∈N. Finally,rdenotes the rank of the2-class group ofQ(√q,√p, i).
110 Frontiers in Science and Engineering - Vol. 6 - n° 1 - 2016
An International Journal Edited by The Hassan II Academy of Science and Technology as capitulation problems, whether class field towers are finite or not and the structures of the2- class groups of the unramified extensions ofk withink2(1). In this paper, we give an example of this situation.
Letk= Q(√pq, i), wherep ≡1 (mod 4)andq ≡3 (mod 4)are two different primes such that
q p
= 1, then the symbols εq
p
:=
εq
pQ(√q)
and
ε2q
p
:=
ε2q
pQ(√2q)
do not depend on the choice ofpQ(√q) andpQ(√2q) (εmis the fundamental unit ofQ(√
m), wherem= qor2q).
According to [1],2εmis a square inQ(√
m)wheneverm=qor2q, so εm
p
= 2
p
. Ifp≡1 (mod 8), then we can define the following quartic residue symbol:
εm p
4
:=
εm pQ(√m,i)
4
(see.
[8]). In the present paper, we give explicit expressions of this symbol form = q and we show that if the2-class group ofkis of type(2,4), then it takes the value−1and we also show that the metacyclicity ofGis characterized by the value of this symbol form= 2q≡14 (mod 16).
Letmbe a square-free integer andk be a number field. Throughout this paper, we adopt the following notations:
• h(m), (resp.h(k)): the2-class number ofQ(√
m), (resp.k).
• εm: the fundamental unit ofQ(√m), ifm >0.
• Ek: the unit group ofOk.
• Wk: the group of roots of unity contained ink.
• ωk: the order ofWk.
• i=√
−1.
• k+: the maximal real subfield ofk, ifkis a CM-field.
• Qk= [Ek :WkEk+]is the Hasse unit index, ifkis a CM-field.
• Cl2(k): the 2-class group ofk.
2. The quartic residue symbol of ε
mand applications
In what follows, we adopt the following notations: ifp ≡ 1 (mod 8)is a prime, then
2 p
4will denote the rational biquadratic symbol which is equal to1or−1, according as2p−14 ≡ 1or −1 (modp). Moreover the symbol p
2
4 is equal to (−1)p−81. A 2-group H is said to be of type (2n1,2n1, ...,2ns)if it is isomorphic toZ/2n1×Z/2n2×...Z/2ns, whereni∈N. Finally,rdenotes the rank of the2-class group ofQ(√q,√p, i).
Frontiers in Science and Engineering
An International Journal Edited by Hassan II Academy of Science and Technology 2 A. Azizi et al. On the quartic residue symbols of certain fundamental units
Theorem 1. Letp =u2∓2v2 ≡1 (mod 8),q = w2−2z2≡3 (mod 4)be primes such that =
2 q
and
q p
= 1. Ifεq is the fundamental unit ofQ(√q), then εq
p
4
= 2
p
4
uz±vw p
. Proof. Asp≡1 (mod 8)andq≡3 (mod 4), then
2
p
= ±2
p = 1;
2 q
=
q 2 q
== 1.
This means thatpandqsplit in the ring of integers ofQ(√
2), so there exist four positive integers u,z,vandwsuch that
p=u2∓2v2 et q =w2−2z2.
Letεq =x+y√qbe the fundamental unite ofQ(√q). Sinceq≡3 (mod 4), so2εqis a square in Q(√q)(see [1]). In this situation, we have that
x+=y21,
x−=qy22, and √
2ε2=y1+y2√q where y = y1y2, then 2 = y12−qy22 and(y1 −√
2ε)(y1+√
2ε) = qy22. Let q = µµ be the decomposition ofqinQ(√
2). Asy1−√
2εis the conjugate ofy1+√
2εinQ(√
2), so we obtain the following decomposition:
y1±√
2ε=µα2, y1∓√
2ε=µα2,
whereα is the conjugate of αandy2 = αα, which implies that 2y1 = µα2+ µα2. Using this equality we can check that
2µ√
2ε2= (µα+α√q)2. Then
2εq pQ(√q)
= 2εq pQ(√q,√2)
=
2µ pQ(√q,√2)
=
2 pQ(√q,√2)
w±√ 2z pQ(√q,√2)
= 2
p
w±√ 2z pQ(√2)
111 On the quartic residue symbols of certain fundamental units A. AZIZI et al.
Frontiers in Science and Engineering - Vol. 6 - n° 1 - 2016
An International Journal Edited by The Hassan II Academy of Science and Technology
A. Azizi et al. On the quartic residue symbols of certain fundamental units
Theorem 1. Letp =u2∓2v2 ≡1 (mod 8),q = w2−2z2 ≡3 (mod 4)be primes such that =
2 q
and
q p
= 1. Ifεq is the fundamental unit ofQ(√q), then εq
p
4
= 2
p
4
uz±vw p
. Proof. Asp≡1 (mod 8)andq≡3 (mod 4), then
2
p
= ±2
p = 1;
2 q
=
q 2 q
== 1. This means thatpandqsplit in the ring of integers ofQ(√
2), so there exist four positive integers u,z,vandwsuch that
p=u2∓2v2 et q =w2−2z2.
Letεq =x+y√qbe the fundamental unite ofQ(√q). Sinceq ≡3 (mod 4), so2εq is a square in Q(√q)(see [1]). In this situation, we have that
x+=y21,
x−=qy22, and √
2ε2=y1+y2√q where y = y1y2, then 2 = y12−qy22 and(y1 −√
2ε)(y1+√
2ε) = qy22. Let q = µµ be the decomposition ofqinQ(√
2). Asy1−√
2εis the conjugate ofy1+√
2εinQ(√
2), so we obtain the following decomposition:
y1±√
2ε=µα2, y1∓√
2ε=µα2,
whereα is the conjugate ofα andy2 = αα, which implies that 2y1 = µα2+µα2. Using this equality we can check that
2µ√
2ε2= (µα+α√ q)2. Then
2εq pQ(√q)
= 2εq pQ(√q,√2)
=
2µ pQ(√q,√2)
=
2 pQ(√q,√
2)
w±√ 2z pQ(√q,√
2)
= 2
p
w±√ 2z pQ(√2)
Frontiers in Science and Engineering
An International Journal Edited by Hassan II Academy of Science and Technology 3
A. Azizi et al. On the quartic residue symbols of certain fundamental units
=
w±√ 2z u±√
2v
=
v u±√
2v
wv±√ 2zv u±√
2v
= v
p
wv±uv u±√
2v
= v
p
wv±uv p
.
According to [7, proposition 5.1, p. 154], we have v
p
= 1, then εq
p
4
= 2
p
4
2εq pQ(√q)
= 2
p
4
uz±vw p
.
Corollary 1. Let k = Q(√pq, i), where p ≡ 1 (mod 8) and q ≡ 3 (mod 4), and εq be the fundamental unit ofQ(√q). If the2-class group ofkis of type(2,4), then
εq p
4
=− 2
p
4
.
Proof. As the2-class group ofkis of type(2,4), so p
q
=− q
p
4
= 1andQk = 1(see [2]).
Recall that Kaplan has proved in [5, theorem B2 andB3] that if Qk = 1, then
uz+vw p
= q
p
4
=−1. This completes the proof of our assertion.
Corollary 2. Letk=Q(√pq, i), wherep≡1 (mod 8)andq ≡3 (mod 4), andrbe the rank of the2-class group ofQ(√q,√p, i). If the2-class group ofkis of type(2,4), then
r=
2, if
2 p
4
=p 2
4; 3, if
p q
= 1and 2
p
4
=−p 2
4.
Proof. In [3], we have shown that, ifq≡3 (mod 4), then, by puttingη= 2εq pQ(√q)
if
p q
= 1,
112
On the quartic residue symbols of certain fundamental units A. AZIZI et al.
Frontiers in Science and Engineering - Vol. 6 - n° 1 - 2016
An International Journal Edited by The Hassan II Academy of Science and Technology
=
w±√ 2z u±√
2v
=
v u±√
2v
wv±√ 2zv u±√
2v
= v
p
wv±uv u±√
2v
= v
p
wv±uv p
.
According to [7, proposition 5.1, p. 154], we have v
p
= 1, then εq
p
4
= 2
p
4
2εq
pQ(√q)
= 2
p
4
uz±vw p
.
Corollary 1. Let k = Q(√pq, i), where p ≡ 1 (mod 8)and q ≡ 3 (mod 4), and εq be the fundamental unit ofQ(√q). If the2-class group ofkis of type(2,4), then
εq p
4
=− 2
p
4
.
Proof. As the2-class group ofkis of type(2,4), so p
q
=− q
p
4
= 1andQk = 1(see [2]).
Recall that Kaplan has proved in [5, theorem B2 andB3] that if Qk = 1, then
uz+vw p
= q
p
4
=−1. This completes the proof of our assertion.
Corollary 2. Letk=Q(√pq, i), wherep≡1 (mod 8)andq ≡3 (mod 4), andrbe the rank of the2-class group ofQ(√q,√p, i). If the2-class group ofkis of type(2,4), then
r=
2, if
2 p
4
=p 2
4; 3, if
p q
= 1and 2
p
4
=−p 2
4.
Proof. In [3], we have shown that, ifq≡3 (mod 4), then, by puttingη= 2εq pQ(√q)
if
p q
= 1,
Frontiers in Science and Engineering
An International Journal Edited by Hassan II Academy of Science and Technology 4 A. Azizi et al. On the quartic residue symbols of certain fundamental units we obtain
r=
1, if
p q
=−1;
2, if p
q
= 1and 2
p
4
=−p 2
4η; 3, if
p q
= 1and 2
p
4
=p 2
4η.
Since the 2-class group of k is of type (2,4), then p
q
= 1 and η = −1 (see the previous corollary).
Lemma 1([6]). Letk = Q(√pq, i), wherep ≡ 1 (mod 1)andq ≡ 3 (mod 4). If q
p
= 1, then there exist an unramified cyclic extension of degree4containingk∗=Q(√p,√q, i)the genus field ofk.
Theorem 2. Letk=Q(√pq, i), wherep≡1 (mod 8)q≡7 (mod 8), and putG= Gal(k(2)2 /k).
Assume the 2-class group of k is of type (2,4), then the group G is metacyclic if and only if ε2q
p
4
=−1.
Proof. As the2-class group of kis of type(2,4), then the extensionk(1)2 /kadmits three abelian subextensions of degree 2 sayKi,2 and three abelian subextensions of degree 4say Ki,4 where i∈ {1,2,3}. The following figure illustrates the situation.
k(2)2
k(1)2
K1,4
K3,4
K2,4
K1,2
K3,2
K2,2
k
Thus [4, Theorem 14, p. 107] yields thatGis metacyclic if and only if the rank of the2-class group of K3,2 is equal to 2, this in turn is equivalent, by Corollary 2 and previous Lemma, to
113 On the quartic residue symbols of certain fundamental units A. AZIZI et al.
Frontiers in Science and Engineering - Vol. 6 - n° 1 - 2016
An International Journal Edited by The Hassan II Academy of Science and Technology
A. Azizi et al. On the quartic residue symbols of certain fundamental units we obtain
r=
1, if
p q
=−1; 2, if
p q
= 1and 2
p
4
=−p 2
4η;
3, if p
q
= 1and 2
p
4
=p 2
4η. Since the 2-class group of k is of type (2,4), then
p q
= 1 and η = −1 (see the previous corollary).
Lemma 1([6]). Letk = Q(√pq, i), wherep ≡ 1 (mod 1)andq ≡ 3 (mod 4). If q
p
= 1, then there exist an unramified cyclic extension of degree4containingk∗=Q(√p,√q, i)the genus field ofk.
Theorem 2. Letk=Q(√pq, i), wherep≡1 (mod 8)q ≡7 (mod 8), and putG= Gal(k(2)2 /k).
Assume the 2-class group of k is of type (2,4), then the group G is metacyclic if and only if ε2q
p
4
=−1.
Proof. As the2-class group ofkis of type(2,4), then the extensionk(1)2 /kadmits three abelian subextensions of degree 2say Ki,2 and three abelian subextensions of degree 4 sayKi,4 where i∈ {1,2,3}. The following figure illustrates the situation.
k(2)2
k(1)2
K1,4
K3,4
K2,4
K1,2
K3,2
K2,2
k
Thus [4, Theorem 14, p. 107] yields thatGis metacyclic if and only if the rank of the2-class group ofK3,2 is equal to2, this in turn is equivalent, by Corollary 2 and previous Lemma, to
Frontiers in Science and Engineering
An International Journal Edited by Hassan II Academy of Science and Technology 5 A. Azizi et al. On the quartic residue symbols of certain fundamental units
2 p
4
p 2
4= 1. Asq≡7 (mod 8), the [8, Theorem 1, p. 690] and the Corollary 2 imply that ε2q
p
4
= εq
p
4
p 2
4=− 2
p
4
p 2
4. This completes the proof of the theorem.
References
[1] A. Azizi, Sur la capitulation des 2-classes d’id´eaux de k = Q(√
2pq, i), o`u p ≡ −q ≡ 1mod4,Acta. Arith.94(2000), 383-399.
[2] A. Azizi et M. Taous, D´etermination des corpsk = Q(√
d, i) dont les2-groupes de classes sont de type(2,4)ou(2,2,2), Rend. Istit. Mat. Univ. Trieste.40(2008), 93-116.
[3] A. Azizi, M. Taous and A. Zekhnini,On the rank of the2-class group ofQ(√p,√q,√
−1), Period. Math. Hungar. Volume 69, Issue 2, (2014), 231-238.
[4] A. Azizi, M. Taous and A. Zekhnini,On the2-groups whose abelianizations are of type(2,4) and applications, Publ. Math. Debrecen.88/1-2(2016), 93-117.
[5] P. Kaplan,Sur le2-groupe de classes d’id´eaux des corps quadratiques.J. Reine angew. Math.
283/284(1976), 313-363.
[6] F. Lemmermeyer,Separants of quadratic extensions of number fields, preprint.
[7] F. Lemmermeyer,Reciprocity Laws, Springer Monographs in Mathematics, Springer-Verlag.
Berlin 2000.
[8] P. A. Leonard and K. S. Williams,The quadratic and quartic character of certain quadratic units II, Rocky Mountain J. Math. (4), 9 (1979), 683-692.
114
On the quartic residue symbols of certain fundamental units A. AZIZI et al.
Frontiers in Science and Engineering - Vol. 6 - n° 1 - 2016
An International Journal Edited by The Hassan II Academy of Science and Technology 2
p
4
p 2
4= 1. Asq ≡7 (mod 8), the [8, Theorem 1, p. 690] and the Corollary 2 imply that ε2q
p
4
= εq
p
4
p 2
4=− 2
p
4
p 2
4. This completes the proof of the theorem.
References
[1] A. Azizi, Sur la capitulation des 2-classes d’id´eaux de k = Q(√2pq, i), o`u p ≡ −q ≡ 1mod4,Acta. Arith.94(2000), 383-399.
[2] A. Azizi et M. Taous, D´etermination des corpsk = Q(√
d, i)dont les 2-groupes de classes sont de type(2,4)ou(2,2,2), Rend. Istit. Mat. Univ. Trieste.40(2008), 93-116.
[3] A. Azizi, M. Taous and A. Zekhnini,On the rank of the2-class group ofQ(√p,√q,√
−1), Period. Math. Hungar. Volume 69, Issue 2, (2014), 231-238.
[4] A. Azizi, M. Taous and A. Zekhnini,On the2-groups whose abelianizations are of type(2,4) and applications, Publ. Math. Debrecen.88/1-2(2016), 93-117.
[5] P. Kaplan,Sur le2-groupe de classes d’id´eaux des corps quadratiques.J. Reine angew. Math.
283/284(1976), 313-363.
[6] F. Lemmermeyer,Separants of quadratic extensions of number fields, preprint.
[7] F. Lemmermeyer,Reciprocity Laws, Springer Monographs in Mathematics, Springer-Verlag.
Berlin 2000.
[8] P. A. Leonard and K. S. Williams,The quadratic and quartic character of certain quadratic units II, Rocky Mountain J. Math. (4), 9 (1979), 683-692.
Frontiers in Science and Engineering
An International Journal Edited by Hassan II Academy of Science and Technology 6
