192 VOLUME 55, NUMBER 5

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1 ON THE -CLASS GROUP OF Q F WHERE F IS A PRIME FIBONACCI NUMBER MOHAMMED TAOUS Abstract Let F be a rime Fibonacci number where > Put k Q F and let k 1 be its Hilbert -class field Denote by k the Hilbert -class field of k 1 and by G Galk /k the Galois grou of k /k In this aer, we characterize the structure of the -class grou of k and we study the metacyclicity of G 1 Introduction Let d be a square-free integer and k Q d a + b d : number field Then we define the ring of integers of k by a, b Q} be a quadratic O k α k : P α 0 for some monic olynomial P Z[X]} Z[ It is known that O k 1+ d ], if d 1 mod ; Z[ d], if not Two ideals I and J of O k are said to be equivalent if I λj for some λ k, this definition of equivalent is an equivalence relation The ideal classes of O k form a finite grou called the class grou of k, and will be denoted by Clk We define the -rank and the -rank of Clk resectively as follows: r dim F Clk/Clk and r dim F Clk /Clk where F is the finite field with elements Several works are interested in determining the structure of the -class grou Cl k, that is the Sylow -subgrou of Clk For examle, for, r and r 0 or 1, we can see the works of P Kalan [13], and Benjamin et all [] As the only erfect squares in the Fibonacci sequence are F 0 0, F 1 F 1 and F 1 1 see, eg, [8], then the quadratic field k Q ±F n is well defined for n / 0, 1,, 1} On the other hand, by genus theory, the -class grou, Cl k, of k Q F n is trivial if and only if F n m where is a rime number Y Kishi [1] gave an infinite family of imaginary quadratic fields Q F n with n mod 0 such that r 1 The latter author and M Aoki gave in [1] another infinite family of airs of imaginary quadratic fields with r 1 Motivated by these works, we thought, in a first time, studying the -class grou of the real quadratic fields k Q F n But we noticed that we cannot do it, in general, since to calculate the rank of Cl k, we must first calculate the rime numbers that divide square-free art of F n To overcome this difficulty, we changed F n by F where F is a rime Fibonacci number with >, and we decided to characterize the structure of the -class grou of k and to study the metacyclicity of G Galk /k where k is the second Hilbert -class field of k 19 VOLUME, NUMBER

2 THE -CLASS GROUP OF Q P F P WHERE F P IS A PRIME FIBONACCI NUMBER Some consequences of the Binet s formula The Fibonacci numbers F n may be defined by the recurrence relation F n F n 1 + F n for n with F 0 0 and F 1 1 In 183, the French mathematician Jacques Philie Marie Binet discovered that F n 1 [ 1 + n 1 n ] 1 This exression of F n is called Binet s formula Using this formula, we can show the well-known identities [1]: n 1 n n 1 F n k Cn k+1 and n F n+1 k Cn+1 k+1 k0 k0 F n+1 F n+1 + F n 3 From which we deduce that all the odd divisors of F n+1 are of the form t + 1 Recall also that the Lucas numbers are the sequence of integers L n n N defined by the linear recurrence equation L n L n 1 + L n for n with L 1 1 and L 3 Theorem 1 Legendre, Lagrange Let be an odd rime integer number F and the Lucas number L have the following roerties: F ±1 1 ± mod, F mod and L 1 Then the Fibonacci mod Those extraordinary sequences have so many other roerties see, eg, [0, 1] We can, for examle, cite the following identities: F n F n L n L n F n 1 n F n L n+1 L n + 1 n 6 Corollary Let be a rime >, and denote by ṗ the Legendre symbol Then the Fibonacci number F has the following roerties: F 1 If 1 mod, then 1 and F F If 3 mod, then F Proosition 3 [1] Let a + b be an odd rime, and suose a odd Then a b a + b 1, and a + b 3 Preliminary In what follows, we adot the following notations: if a 1 and 1 mod, then a will denote the rational biquadratic symbol which is equal to 1 or 1, according as a 1 1 or 1 mod, in articular 1 1 DECEMBER

3 THE FIBONACCI QUARTERLY Lemma 31 [17] If > is a rime such that 1 mod, then F F + and 1 0 mod Proof From the first equality of Theorem 1, we conclude that According to Formula, we have F 0 mod and F + 1 mod 31 F F L 0 mod We now show that never divides L If 1, then Formula imlies that L +1 F +1 1 This is absurd 1 +1 mod If divides L, then, since 1 mod, If 1, then by relacing, in 6, n by 1, we get F 1 L L 1 + According to the third equality of Theorem 1 and to the first equality of 31, we have L 1 mod, ie, never divides L Then F 0 mod Finally, from the second equality of Theorem 1 and 3, we can see that As is rime, so F + F F + ± 1 + F 1 F + mod mod This gives that mod F + 1 Theorem 3 Burde, [7] Let m, n N be odd such that m a +b and n c +d, where a, b, c, d N, ac, and a, b c, d m, n 1 Suose that m n n m 1 Then m n n m ac + bd ac + bd n m Proosition 33 Let F be a Fibonacci number with rime index 1 mod Then we have F 1, if 1 mod 3; F, if not Proof Let a + b be an odd rime, and suose that a is odd and ut 3 1 i It is easy to see that + 3 mod 6, this imlies that 3 is not divisible by 3, so F + 3 is 19 VOLUME, NUMBER

4 THE -CLASS GROUP OF Q P F P WHERE F P IS A PRIME FIBONACCI NUMBER odd Since F F F 3, then the revious theorem gives that F F af + 3 a b + bf 3 F + 3 F 3, if 3 ;, if see the lemma i, if 3 ; 1 1 i, if i 3 Proosition 3 To show the following results, it suffices to use Formula modulo Lemma 3 Let F be a Fibonacci number with rime index 1 mod Then 1 F mod F 1 1 mod 3 F mod If F 1, then Corollary 3 Let F be a Fibonacci number with rime index 1 mod If 1, F then F 3 DECEMBER

5 THE FIBONACCI QUARTERLY Proof Since 1 +, F + 3 that F F is odd and F F + 3 F F , if 3 1; 1, if F 3, then Theorem 3 imlies, if 3 1;, if 3 1 Main results Let k be an algebraic number field and let Cl k denote its -class grou Denote by k 1 the Hilbert -class field of k and by k its second Hilbert -class field Put G Galk /k and denote by G its derived grou; then it is well known, by class field theory, that G/G Cl k x, y For any rime l, l k will denote a rime ideal of k lies above l We also denote by l k x res the Hilbert symbol res the quadratic residue symbol for the rime l k alied l k to x, y res x Recall that a -grou H is said to be of tye n 1, n 1,, ns if it is isomorhic to Z/ n 1 Z Z/ n Z Z/ ns Z, where n i N If k Q F such that F is a rime Fibonacci number where >, then, by genus theory, rankcl k Thus Cl k is of tye n, m with n, m N Hence grou theory imlies that Cl k admits three normal subgrous of index, denote them by H i, i 1,, 3} The following diagram illustrates the situation : Cl k H 1 H 1 H 3 On the other hand, by class field theory, each subgrou H i of Cl k is associated to a unique unramified extension F i within k 1 such that H i /H i Cl F i The situation is reresented 196 VOLUME, NUMBER

6 THE -CLASS GROUP OF Q P F P WHERE F P IS A PRIME FIBONACCI NUMBER by the following figure: k F 1 F k 1 1 F 3 According to [18, Theorem ], the three fields F i are given as follows: F 1 Q, F, F Q, F and F 3 Q F, Recall also that a grou G is said to be metacyclic if it has a normal cyclic subgrou H such that the quotient grou G/H is cyclic For examle, if Cl k is of tye,, then G is metacyclic More recisely and by [19], G is isomorhic to one of the following grous: 1 Abelian -grou of tye, The dihedral grou 3 The quaternion grou The semidihedral grou Theorem 1 Main result: case 1 Let F be a rime Fibonacci number such that > and 1 Put k Q F and let k 1 be its Hilbert -class field Denote by k the Hilbert -class field of k 1 and by G Galk /k the Galois grou of k /k Then 1 Cl k is of tye, G is abelian is of tye, if and only if mod or 3 mod and F 1 mod 8 3 If G is nonabelian, then it is quaternion, dihedral or semi-dihedral Proof If 1 mod, then Corollary imlies that F 1, so we use Kalan s results on the -class grou of the field Q 1 3 where i 1 mod [13] to show that Cl k is of tye, In the case where 3 mod, we use a result of E Benjamin and C Snyder namely [, case 7, age 163] to deduce that Cl k is also of tye, According to [], G is abelian if and only if 1 mod and F F 1 or 3 mod and F 1 mod 8 This is equivalent, by Proosition 33 and the Chinese remainder theorem, to mod or 3 mod and F 1 mod 8 Lemma Let F be a Fibonacci number with rime index 1 mod Denote by r i the rank of the -class grou of the field F i, where i 1,, 3} Assume 1 if mod 8 or 1 If mod 3, then r 1 r 3 and r 1; 3, if not if If 1 mod 3, then r 3 3 and r 1 r 1; 3, if not Proof This is an immediate consequence of Proosition 33, Corollary 3 and Theorem of [3] DECEMBER

7 THE FIBONACCI QUARTERLY Lemma 3 Let F be a Fibonacci number with rime index 3 mod Denote by r i the rank of the -class grou of the field F i, where i 1,, 3} If 1, then r1 r, if F mod 8; and r 3 3, if F 1 mod 8 Proof As the class number of F Q is odd, then the ambiguous class number formula see [9] imlies that r t e 1, where t is the number of rimes of F that ramify in F /F and e is defined by e [E F : E F N F /F k ] The Hasse norm theorem see, eg, [11, theorem 6, 179] imlies that a unit ε of F is a norm of an element of F F F F, ε if and only if 1, for all l F F rime ideal of F Denote by ε the fundamental l F unit of F Q, so E F, the unit grou of F, is equal to 1, ε According to [], ε is a square in F, then F, ε ε ε 1, Q Q P F, 1 1 l 1, if l or l F, l F 1, if not, if F mod 8, Thus we conclude that e 1 and r To rove r 1 and r 3 3, if F 1 mod 8, we can aly Theorem 1 of [3] Theorem Main result: case 1 Let F be a rime Fibonacci number such that > and 1 Put k Q F and let k 1 be its Hilbert -class field Denote by k the Hilbert -class field of k 1 and by G Galk /k the Galois grou of k /k Then 1 If 3 mod, then Cl k is of tye, n, such that n and G is metacyclic if and only if F mod 8 If 1 mod, then G is metacyclic if and only if mod or [ mod 3 and 1] Proof 1 If 3 mod, then the last lemma and Theorem 1 of [6] yield that G is metacyclic if and only if F mod 8 To show that Cl k is of tye, n, it suffices to rove that r, the -rank of Cl k, is 1 In this case, the discriminant of k is 0F, and the only ossible C-decomosition of is 1 with 1 F and 0, if F 1 mod 8; 1 0 and F, if not According to [16], r equals the number of indeendent C-decomositions of, so r 1 If 1 mod, then we just aly Lemma and Theorem 1 of [6] Numerical examles with all known F To date, F is known to be rime for 3,,, 7, 11, 13, 17, 3, 9, 3, 7, 83, 131, 137, 39, 31, 33, 9, 09, 69, 71, 971, 73, 387, 9311, 9677, 131, 61, 3077, 3999, 3711, 0833, In addition to these roven Fibonacci rimes, there have been found robable rimes for 10911, 13001, 18091, 01107, , 33781, 9001, 93689, 60711, 93117, , 18607, , , , 9033 See[1] Using the Pari software, [10], we find that there are u to now rimes Fibonacci numbers such that 198 VOLUME, NUMBER

8 THE -CLASS GROUP OF Q P F P WHERE F P IS A PRIME FIBONACCI NUMBER G is nonmetacyclic We use the following abbreviations: M means metacyclic, NM means nonmetacyclic Cl k G mod Cl k G mod 7 3-1, M ,? M , NM ,? NM , M , M , M ,? NM 3 3-1, M ,? NM 9 1 1, M , M 3 3-1, M ,? NM 7 3-1, M ,? M , M ,? NM , NM ,? NM , M , M ,? NM ,? NM ,? NM ,? M , M ,? NM 9 1 1,? M ?,? NM ,? M ,? M ,? M ,, ,? M , M ,? M , M , M , M , M ,? NM ,? NM ?,? NM ,, , M Figure 1 Numerical examles References [1] M Aoki and Y Kishi, An infinite family of airs of imaginary quadratic fields with both class numbers divisible by five, submitted [] A Azizi, Sur la caitulation des -classes d idéaux de k Q q, i, où q 1 mod, Acta Arith 9 000, [3] A Azizi and A Mouhib, Sur le rang du -groue de Q m, d où m ou bien un remier 1 mod, Trans Amer Math Soc , 71-7 [] E Benjamin and C Snyder, Real quadratic number fields with -class grou of tye,, Math Scand , [] E Benjamin, F Lemmermeyer and C Snyder, Real quadratic fields with abelian -class field tower, J Number Theory , [6] N Blackburn, Generalizations of certain elementary theorems on -grous, Proc London Math Soc ,1- [7] K Burde, Ein rationales biquadratisches Rezirozitätsgesetz, J Reine Angew Math , [8] J H E Cohn, On square Fibonacci numbers, J London Math Soc , 37-0 [9] C Chevalley, Sur la théorie du cors de classes dans les cors finis and les cors locaux, J Fac Sc Tokyo, Sect 1, t, 1933, [10] The PARI Grou, PARI/GP, Bordeaux, Version 7, Oct 6 01, htt://arimathu-bordeauxfr [11] G Gras, Class field theory, from theory to ractice, Sringer Verlag 003 DECEMBER

9 THE FIBONACCI QUARTERLY [1] R P Grimaldi, Fibonacci and Catalan numbers An introduction, John Wiley & Sons, Inc, Hoboken, NJ, 01 [13] P Kalan, Sur le -groue de classes d idéaux des cors quadratiques, J Reine angew Math 83/8 1976, [1] Y Kishi, A new family of imaginary quadratic fields whose class number is divisible by five, J Number Theory , 08 [1] F Lemmermeyer, Recirocity Laws, Sringer Monograhs in Mathematics, Sringer-Verlag Berlin 000 [16] L Rédei, Die Anzahl der durch teilbaren Invarianten der Klassengrue eines beliebigen quadratischen Zahlkörers, Math Naturwiss Anz Ungar Akad d Wiss 9 193, [17] Z H Sun and Z W Sun, Fibonacci numbers and Fermat s last theorem, Acta Arith , [18] B K Searman, K S Williams, Unramified Quadratic Extensions of a Quadratic Field, Rocky Mountain J Math 199, no, [19] O Taussky, A Remark on the class field tower, J London Math Soc , 8-8 [0] N N Vorobiev, Fibonacci numbers, Russian Academy of Sciences, 199 6th edition Nauka, Moscow [1] Wikiedia, Fibonacci rime, htts://enwikiediaorg/wiki/fibonacci_rime#cite_note-rto- Deartment of Mathematics, Faculty of Sciences and Technology, Moulay Ismaïl University, Errachidia, Morocco address: taousm@hotmailcom 00 VOLUME, NUMBER