Gulf Journal of Mathematics Vol 4, Issue 4 (2016) 166-170
UNITS OF SOME ABELIAN IMAGINARY NUMBER FIELDS OF TYPE (2, 4) A. AZIZI, I. JERRARI, A. ZEKHNINI, AND M. TALBI Abstract. Let K be an abelian imaginary number field of type (2, 4), i.e. the Galois goup Gal(K/Q) ' Z/2Z × Z/4Z. In this paper, we determine the fundamental system of units of some field K.
1. Notations Let m be an integer and K be a number field. Throughout this paper, we adopt the following notations: • EK : the unit group of K; • K + : the maximal real subfield of K, if it is a CM-field; • QK + = [EK : WK EK + ] is Hasse’s unit index, if K is a CM-field; • p K : the prime ideal of K; x, y x • (resp. ): the Hilbert symbol (resp. the quadratic residue pK pK symbol) for the prime pK applied to (x, y) (resp. x); • ξm : a primitive m-th root of unity; • Q(ξm ): the m-th cyclotomic number field. 2. Units of some abelian imaginary number fields of type (2, 4) By using the results of M. N. Gras [5], we will define a fundamental system of units (FSU) of a real cyclic quartic number field. Let L be a real cyclic quartic number field of Galois group H = hσL i and of quadratic subfield k, and let ε be the fundamental unit of k. Let fL (resp. fk ) be the conductor of L (resp. k); then fL = fk · g where g is an integer, and the discriminant of L is equal to disc(L) = fk · fL2 . Since fL is the conductor of L, then we have L ⊂ Q(ξfL ). 2 Let χL be a rational character of Q(ξfL ), EχL = {ω ∈ EL : ω 1+σL = ±1}, |EL | (resp. |Ek |, |EχL |) be the group of the absolute values of EL (resp. Ek , EχL ), |E L | = |Ek | ⊕ |EχL |, QL = [|EL | : |E L |] and εχL be a generator of EχL . From [5], we have QL = 1 or 2, which gives two possible structures for the unit group of L: Date: Accepted: Oct 24, 2016. ∗ Corresponding author. 2010 Mathematics Subject Classification. 11R16, 11R11, 11R29. Key words and phrases. Unit Group, Minkowski’s Unit, Hasse’s Unit Index. This work is partially supported by CNRST (PBER), Hassan II Academy of Sciences and Technology (Morocco), URAC6 and ACSA laboratory (FSO-UMPO). 166
UNITS OF SOME ABELIAN IMAGINARY NUMBER FIELDS OF TYPE (2, 4)
(1) If QL = 1, then we have EL = h−1, ε, εχL , εσχLL i;
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σ2
L (2) If QL = 2, then we have EL = h−1, ξL , ξLσL , ξLL i with ξL2 = ±εεχ1−σ , L
1+σ 2
ξL1+σL = ±εχL and ξL L = ±ε. Note that ξL is called a Minkowski’s unit. √ l) and deProposition 2.1. Let l ≡ 1 (mod 4) be a prime number. Put k = Q( p √ note by ε its fundamental unit. Then the field L = k( ε l) admits a Minkowski’s unit if and only if l ≡ 1 (mod 8). Proof. It is known that QL = 2 if and only if L admits a Minkowski’s unit, and from [3, Th´eor`eme II.2], the field L admits a Minkowski’s unit if and only if the unit ε is norm in L. Let p be a prime ideal of k, so we have: √ • If p is not above l, then vp (ε l) = 0, so; √ vp (ε√l) ε, ε l ε = p p = 1. √ √ • If p lies above l, then vp (ε l) = 1. Put ε = x + y l, then x = 2j c with c odd and j ≥ 1, so; √ vp (ε√l) ε, ε l ε = p p ε = p x = pj 2c = pj 2c = lj 2 c = l j l 2 l = l c Since x2 − ly 2 = 4j c2 − ly 2 = −1, then ly 2 = 4j c2 + 1 ≡ 1 (mod c). Thus, √ j 2 ε, ε l = p l 2 If l ≡ 5 (mod 8) then j = 1 and = −1, and if l ≡ 1 (mod 8) then l 2 j > 1 and = 1. l
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This means that the unit ε is not norm in L in the first case, whereas in the second case ε is. So the result. √ Corollary 2.2. Let l ≡ 1 (mod 4) be a prime p number. Put k = Q( l) and √ denote by ε its fundamental unit, and set L = k( ε l). Then: (1) If l ≡ 5 (mod 8), we have QL = 1; (2) If l ≡ 1 (mod 8), we have QL = 2. Remark 2.3. In [2], the authors have shown 2 in another way. Lemma 2.4. Let p ≡ 3 (mod 4) and l ≡ 1 (mod 4) be differentqprimes. Put √ √ k = Q( l) and denote by ε its fundamental unit. Let L = k( pε l), then QL = 1. Proof. It is well known that p divide the conductor fL of field L and the norm of the unit ε is equal to −1, and so, from [5, Proposition 4], we have QL = 1. Let √ M0 be a totally real number field, β be a positive number M0 , M = M0 ( −β) be a quadratic extension of M0 , absolutely abelian and finite. It is known that Hasse’s unit index QM + is equal to 1 or 2. Which means that a FSU of M0 is also a FSU of M in the first case, whereas both systems different in the second case. From [1], we have the following results: Let j be an integer greater than or equal to 2 and ξ2j be a primitive 2j -th root of unity, then there exist two real numbers µj and λj defined inductively: p √ √ ξ2n = 21 (µn + λn −1), where µn = 2 + µn−1 , λn = 2 − λn−1 √ µ2 = 0, λ2 = 2 and µ3 = λ3 = 2. Let j0 be the greatest integer such that ξ2j0 is contained in M .
√ Proposition 2.5 ([1]). Let M0 be a totally real number field, M = M0 ( −1) be a quadratic extension of M0 , absolutely abelian and finite, and {ε1 , ε2 , . . . , εr } be a FSU of M0 (whose the units are all positive). So we have: jr−1 • If there exists a unit λ of M0 of the form εj11 εj22 . . . εr−1 εr (up to a permutation), where jk√∈ {0, 1}, such that (2 + µj0 )λ is a square in M0 , then {ε1 , ε2 , . . . , εr−1 , ξ2j0 λ} is a FSU of M . • Otherwise, {ε1 , ε2 , . . . , εr } is a FSU of M . Proposition 2.6 ([1]). Let M0 be a totally real number field, √ √ β be a positive number of M0 , square free and such that β ∈ / M0 , M = M0 ( −β) be a quadratic extension of M0 , absolutely abelian and finite, and {ε1 , ε2 , . . . , εr } be a FSU of M0 . Assume that the units εj are positive. So we have: j
r−1 • If there exists a unit λ of M0 of the form εj11 εj22 . . . εr−1 εr (up to a permutation), where√ jk ∈ {0, 1}, such that βλ is a square in M0 , then {ε1 , ε2 , . . . , εr−1 , −λ} is a FSU of M . • Otherwise, {ε1 , ε2 , . . . , εr } is a FSU of M .
Lemma 2.7. Let p ≡ 3 (mod 4) and l ≡ 5 (mod 8) be different Put q √ primes. √ √ k = Q( l) and denote by ε its fundamental unit. Let F = k( pε l, −1), then QF + = 1.
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Proof. From Proposition 2.5, to show that the index QF + = 1, it suffices to show that (2 + µ2 )λ = 2λ is not a square in F + . In fact, suppose that 2λ is a square in F + , which equivalent to (2)F + = H2 where H is a ideal of F + , which is not the case, since 2 cannot ramify in F + . Lemma 2.8. Let p ≡ 3 (mod 4) and l ≡ 1 (mod 4) be different Put p √ primes. √ √ 0 k = Q( l) and denote by ε its fundamental unit. Let F = k( ε l, −p), then QF 0+ = 1. Proof. From Proposition 2.6, to show that the index QF 0+ = 1, it suffices to show that pλ is not a square in F 0+ . In fact, suppose that pλ is a square in F 0+ , which equivalent to (p)F 0+ = H2 where H is a ideal of F 0+ , which is impossible, since p cannot ramify in F 0+ . Corollary 2.2, and Lemmas 2.4, 2.7 and 2.8 allow us to deduce the following theorem: Theorem 2.9. Let p ≡ 3 (mod 4) and l ≡ 1 (mod 4) be different Put q primes. √ √ k = Q( l) and denote by ε its fundamental unit, and set K = k( −pε l). Let √ √ F = K( −1) and F 0 = K( −p), then (1) If l ≡ 5 (mod √ 8), we have:σ • EF = h −1, ε, εχF + , εχFF ++ i; σ h−1, ε, εχF 0+ , εχFF 0+ if p 6= 3; 0+ i • EF 0 = σF 0+ hξ6 , ε, εχF 0+ , εχF 0+ i if p = 3. (2) If l ≡ 1 (mod 8), we have ( σ2 σ h−1, ξF 0+ , ξFF0+0+ , ξFF0+0+ i if p 6= 3; EF 0 = σ 2 0+ σF 0+ hξ6 , ξF 0+ , ξF 0+ , ξFF0+ i if p = 3. 3. ACKNOWLEDGMENT The authors would like to thank Sidi Mohamed Ben Abdellah Univercity (USMBA), LSI and FP of Taza in MOROCCO for their valued supports. References 1. A. Azizi, Unit´es de certains corps de nombres imaginaires et ab´eliens sur Q, Ann. Sci. Math. Qu´ebec, 23 (2), 15-21, (1999). 2. A. Azizi, M. Talbi and M. Talbi, Sur la tour de Hilbert de certains corps, Bol. Soc. Paran. Mat. (3s.) 34 2 (2016), 107-112. 3. L. Bouvier and J. J. Payan, Modules sur certains anneaux de Dedekind. Application ` a la structure du groupe des classes et ` a l’existence d’unit´es de Minkowski, J. Reine Angew. Math. 274/275 (1975), 278-286. 4. E. Brown and C. J. Parry, The 2-class group of certain biquadratic number fields I, J. Reine Angew. Math. 295 (1977), 61-71. 5. M. N. Gras, Table num´erique du nombre de classes et des unit´es des extensions cycliques r´eelles de degr´e 4 de Q, Publ. Math. Fac. Sciences de Besancon, Th´eorie des Nombres (1977-78).
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Abdelmalek Azizi: Mohammed First University, Department of Mathematics, Faculty of Sciences, Oujda, Morocco. E-mail address:
[email protected] Idriss Jerrari: Mohammed First University, Department of Mathematics, Faculty of Sciences, Oujda, Morocco. E-mail address: Idriss
[email protected] Abdelkader Zekhnini: Mohammed First University, Department of Mathematics and Informatics, Pluridisciplinary Faculty of Nador, Morocco. E-mail address:
[email protected] Mohammed Talbi: Regional Center of Education and Training, Oujda, Morocco. E-mail address:
[email protected]
