A generalization of Scholz s reciprocity law

Transcription

1 Journal de Théorie des Nombres de Bordeaux , A generalization of Scholz s recirocity law ar Mark BUDDEN, Jeremiah EISENMENGER et Jonathan KISH Résumé. Nous donnons une généralisation de la loi de récirocité de Scholz fondée sur les sous-cors K t 1 et K t de Qζ de degrés t 1 et t sur Q, resectivement. La démonstration utilise un choix articulier d élément rimitif our K t sur K t 1 et est basée sur la division du olynôme cyclotomiue Φ x sur les sous-cors. Abstract. We rovide a generalization of Scholz s recirocity law using the subfields K t 1 and K t of Qζ, of degrees t 1 and t over Q, resectively. The roof reuires a articular choice of rimitive element for K t over K t 1 and is based uon the slitting of the cyclotomic olynomial Φ x over the subfields. 1. Introduction In 193, Scholz [1] roved a rational uartic recirocity law via class field theory. While the law still bears Scholz s name, it was recently noted by Lemmermeyer see the notes at the end of Chater 5 in [11] that it had been roved much earlier in 1839 by Schönemann [13]. Since then, Scholz s recirocity law has been roved using many different methods see [3], [7], [10], and [1] for other roofs. The unfamiliar reader is referred to Emma Lehmer s exository article [9] for an overview of rational recirocity laws and Williams, Hardy, and Friesen s article [15] for a roof of an all-encomassing rational uartic recirocity law that was subseuently simlified by Evans [] and Lemmermeyer [10]. We begin by stating Scholz s recirocity law and an octic version of the law roved by Buell and Williams []. We will need the following notations. For a uadratic field extension Q d of Q, with suarefree ositive d Z, let ε d denote the fundamental unit and hd denote the class number. The standard notation will be used to denote the Legendre symbol. We will also need to define the rational ower residue symbol. Assume that a is an integer such that a, 1 that satisfies Manuscrit reçu le 1 juillet 006. a 1 n 1 mod

2 58 Mark Budden, Jeremiah Eisenmenger, Jonathan Kish for a rational rime and a ositive integer n. Then define the rational symbol a a 1 n mod. n This symbol takes on the same values as a, the Qζ n nth ower residue symbol where is any rime above in Qζ n. For our uroses, n will usually be a ower of. It should also be noted that the Legendre symbol is euivalent to our rational ower residue symbol when n 1. By convention, we define a 1 for all a such that a, 1. 1 Theorem 1.1 Scholz s Recirocity Law. Let 1 mod be distinct rational rimes such that 1. Then ε ε. In [1], Buell and Williams conjectured, and in [] they roved, an octic recirocity law of Scholz-tye which we refer to below as Buell and Williams recirocity law. Although their law is more comlicated to state, it does rovide insight into the otential formulation of a general rational recirocity law of Scholz-tye. Theorem 1. Buell and Williams Recirocity Law. Let 1 mod 8 be distinct rational rimes such that 1. Then ε ε if Nε h/ ε if Nε 1 ε where N is the norm ma for the extension Q over Q. Buell and Williams succeed in roviding a rational octic recirocity law involving the fundamental units of uadratic fields, but it loses some of the simlicity of the statement of Scholz s recirocity law and reuires the introduction of class numbers. It seems more natural to use units from the uniue uartic subfield of Qζ when constructing such an octic law. This was our motivation in the formulation of a general rational recirocity law similar to that of Scholz. In Section, we describe a rimitive element for the uniue subfield K t of Qζ satisfying [K t : Q] t, when 1 mod t and t 1. Our choice of a rimitive element involves a secific choice of a unit η t O K. Section 3 rovides the roof of a generalization of Scholz s t 1 recirocity law after giving a thorough descrition of the rational residue

3 Generalization of Scholz s law 585 symbols used in the theorem. We show that whenever 1 mod t are distinct rimes with t and 1, t 1 t 1 t then where β t t k t t β t t 1 η k k O K t 1 and O K t 1 is any rime above. Our roof is based uon the slitting of the cyclotomic olynomial Φ x over the fields in uestion. In the secial case where t, we note that η ε, resulting in the statement of Scholz s recirocity law. Since our rational recirocity law takes on a simler octic form than Buell and Williams recirocity law, comaring the two results in an interesting corollary. It is observed that if 1 mod 8 are distinct rimes satisfying 1, then η8 ε Nε h/, where O K is any rime above. Acknowledgements The work contained here was artially suorted by two internal grants from Armstrong Atlantic State University. The authors would like to thank Sungkon Chang for carefully reading a reliminary draft of this manuscrit and roviding comments which cleaned u some of our arguments and fixed an oversight in Section 3. Thanks are also due to the referee for several comments that imroved the exosition.. Subfields of Qζ When is an odd rational rime, it is well-known that the uniue uadratic subfield of Qζ is K Q where 1 1/. In this section, we rovide a useful descrition of the subfield K t of Qζ of degree t over Q when 1 mod t and 1. We will need the t following variant of Gauss s Lemma that is due to Emma Lehmer see [8] or Proosition 5.10 of [11] which we state without roof. Lemma.1. Let l and mn+1 be rational rimes such that l 1 n and let A {α 1, α,..., α m }

4 586 Mark Budden, Jeremiah Eisenmenger, Jonathan Kish be a half-system of n th ower residues. Then lα j 1 aj α πj mod for some ermutation π of {1,,..., m} and l 1 µ where µ n m ai. n should be noted that Lemma.1 can be alied to the evaluation of l It for all 1 l < by using Dirichlet s theorem on arithmetic rogressions and the observation that the rational residue symbol is well-defined on congruence classes modulo. The statement of Scholz s Recirocity Law utilizes the fundamental unit of K. Our general rational recirocity law will similarly reuire the units described in the following theorem. Theorem.. Let 1 mod t be a rational rime with t such that 1 and set t { A t 1 a 1 } a 1 t 1 and { B t 1 b 1 } b b 1 and 1. t t 1 Then the element ζ b ζ b b B η t t ζ a ζ a a A t is a unit in O. K t 1 Proof. Let 1 mod t be a rime such that 1 ie., t A t B t and suose that η t is defined as in the statement of the theorem. We begin by noting that ζ Qζ. This is easily checked by observing that Qζ Qζ and that both fields have degree 1 over Q. One can check that η t Z[ζ ] by comuting the norm of η t in Qζ and noting that A t and B t have the same cardinality. Next, we show that η t O K t 1. To do this, we define the element ζ b ζ b b B η t t ζ a ζ a. a A t i1

5 Generalization of Scholz s law 587 If 1, then a ±a mod, b ±b mod, and t 1 ζ b ζ b ζ b ζ b ζ b ζ b η t b B t a A t ζ a ζ a b B t a A t ζ a ζ a b B t a A t η t. ζ a ζ a Alying a similar argument to the case 1, we have that t 1 { η t if A η t t η 1 if B t t. Next, we show that η t K t 1 by showing that it is fixed under all automorhisms σ r GalQζ /Q with r Z/Z t 1. For such residues, we have ra ±a mod and rb ±b mod, which gives σ r η t b B t a A t ζ rb ζ ra ζ rb ζ ra 1 µ B t +µ A t b B t a A t ζ b ζ b ζ a ζ a where µ A t resectively, µ B t counts the number of negatives resulting from ra a mod resectively, rb b mod. By Lemma.1, it follows that r σ r η t η t. t Thus, we see that η t K t 1 Z[ζ ] O K t 1. Similarly, it can be shown that η 1 K t t 1 Z[ζ ] O K t 1 and we conclude that η t, η 1 K t t 1 Z[ζ ] O K t 1, resulting in the claim of the theorem. The descrition of η and the fact that K Q η 1 1/ when 1 mod was shown in Proosition 5.9 and the discussion in Section 3. of Lemmermeyer s book [11], where it was subseuently used to rove Scholz s Recirocity Law. The following theorem includes a roof of this claim along with its extension to describe the subfield K t when t 3. Theorem.3. If 1 mod t is rime with t and then K t Qα t, where α t η 1/ t η 1/ t 1 η 1/t 1 1 1/t+1 t 1 1/ t, 1, t

6 588 Mark Budden, Jeremiah Eisenmenger, Jonathan Kish and K t is the uniue subfield of Qζ satisfying [K t : Q] t. Proof. We begin with the cases t, 3 then roceed by induction on t. Define N t ζ b ζ b and R t ζ a ζ a b B t a A t so that η t N t R for all t. First consider the case when 1 mod t and N R ζ b ζ b ζ a ζ a b B a A 1 1 ζ k ζ k ζ1 k ζ k k1 k1 1 1/ N Qζ/Q1 ζ. Then the sign of N R is 1 1/ resulting in Substituting N η 1 R, we have N R 1 1/ 1/. N η 1 1/ 1/ N η 1/ 1 1/8 1/. Since N K, but N Qζ, and using the fact that Qζ is a cyclic extension of Q, we see that K K N QN. Now for the t 3 case we assume 1 mod 8, in which case we have 1. Then N R N N8 R8, N8 R8 η 1 1/ 1/ 1 η 1 1 1/ 1/, and N 8 R 8 η 1/ 1 1/8 1/. Using η 8 N 8 R 8, we have and N 8 η 8 η 1/ 1 1/8 1/, N 8 η 1/ 8 η 1/ 1 1/16 1/8. We handle the remaining cases by induction on t 3. Suose that for k > 3, 1 mod k, 1, and k N k 1 η 1/ k 1 η 1/ k η 1/k 1 1/k 1/k 1

7 is a rimitive element for K k 1. Using η k 1 R k 1 N kr k N k η 1 k, we have and Generalization of Scholz s law 589 N k η kr k 1 η kn k 1η 1 k 1, N k 1 R k 1, η k N k η kη 1/ k 1 η 1/k 1 1/k 1/k 1, N k η 1/ k η 1/ k 1 η 1/k 1 1 1/k+1 1/k. Again alying the fact that Gal Qζ /Q is cyclic we have that K k K k 1N k QN k N k R k, and is the uniue subfield of Qζ with [K k : Q] k. For our uroses, we will need the following descrition of K t. Note that the element t α t j is also a rimitive element for K t over K t 1 and a simle inductive argument shows that N j α t η 1/ t η 1/ t 1 η 1/t 1 1 1/t+1 t 1 1/ t for all t. 3. Generalized Scholz-tye recirocity law Before describing the main result, we need to exlain the meaning of the symbol η whenever 1 mod t, t 1, η O t 1 K, and t 1 O is any rime above. In this case, slits comletely in O K t 1 K t 1 and we have O /O K t 1 K t 1 Z/Z. Recall that if γ, δ O K t 1, we write γ δ mod if and only if divides γ δ see Chater 9, Section of [5]. While this definition makes sense for γ and δ in the ring of integers of integers of any extension field of K t 1, every element of O K t 1 can be identified with a uniue element from the set {0, 1,..., 1} since its elements are incongruent modulo and may therefore be used as coset reresentatives in O K t 1 /O K t 1.

8 590 Mark Budden, Jeremiah Eisenmenger, Jonathan Kish Let a {0, 1,..., 1} be the uniue element which deends uon the choice of such that Whenever we have then one can define η η a mod. a 1/t 1 1 mod, : a a 1/t t mod. t Of course, the symbol η is only defined when η t 1. This symbol is well-defined, but we must not forget its deendence t 1 on. Next, we state our rational recirocity law using the rational symbols defined above. The roof of the recirocity law deends uon the slitting of the cyclotomic olynomial Φ x over the subfields K t 1 and K t and is modelled after the roof of uadratic recirocity given after Proosition 3. in Lemmermeyer s book [11]. Theorem 3.1. If 1 mod t are distinct odd rimes with t and 1, t 1 t 1 t then where β t t k t β t t η k k O K t 1 and O K t 1 t 1 is any rime above. Proof. Let and be rimes satisfying the hyotheses of Theorem 3.1. The minimal olynomial of ζ over K t 1 is given by ϕx x ζ r O [x], K t 1 r R t 1 where { } R t 1 1 r 1 r 1. t 1 Factoring ϕx over O K t, we obtain ϕx ψ 1 xψ x where ψ 1 x x ζ r and ψ x x ζ n, r R t n N t R t is defined analogously to R t 1, and N t R t 1 R t. Also define the olynomial ϑx ψ 1 x ψ x O K t [x].

9 Generalization of Scholz s law 591 Let σ denote the nontrivial automorhism in GalK t/k t 1, and note that σ σ m K t where σ m GalQζ /K t 1 GalQζ /Q is the automorhism σ m ζ ζ m with m N t. It follows that and since K t K t 1α t, we have σϑx ϑx σα tϑx α tϑx O K t 1 [x]. Thus, it is ossible to write ϑx α tφx for some φx O [x]. K t 1 Following the discussion before Theorem 3.1, let denote any rime above in O K t 1. Consider the congruence 3.1 ϑx ψ 1 x ψ x ϑx mod, t which is actually defined on the ring O K t. On the other hand, we have ϑx α t φx mod. By an analogue of Fermat s little theorem, whenever κ O K t 1, κ κ N 1 κ κ mod, where the norm ma N is the norm of the field extension K t 1 over Q. Since φx O K t 1 [x], we have 3. ϑx α 1 t α tφx mod α t t 1/t ϑx mod. Comaring 3.1 and 3. gives 3.3 ϑx α t t 1/t ϑx mod. t Next, we show that ϑx ψ 1 X ψ X 0 mod. By Kummer s Theorem [6], Theorem 7., the ideal generated by in Z[ζ ] decomoses in exactly the same way as Φ X decomoses in Z/Z[X]. Since and are distinct rimes, the ideal generated by in Z[ζ ] is unramified. If ϕx ψ 1 X mod,

10 59 Mark Budden, Jeremiah Eisenmenger, Jonathan Kish then we can ick {0, 1,... 1} as coset reresentatives of O /O K t 1 K t 1 Z/Z to obtain a suare factor of Φ X in Z/Z[X], contradicting the observation that does not ramify in Z[ζ ]. Thus, 3.3 simlifies to 3. t α t t 1/t mod, and we note that α t O K t 1, which can therefore be identified with an element in {0, 1,..., 1}. The roof is comleted by induction on t. If t, we have η α 1/ mod. Both residue symbols only take on the values ±1 so that we have η η η since is indeendent of the choice of. Now suose that the theorem holds for t k 1, 1 mod k, and k 1 k 1 1. k In articular, we have k 1 Then 3. gives α k k 1/k k Again, the value of k 1 β k β k 1 β k k k 1. k 1 k mod. is indeendent of the choice of and all of the residue symbols only take k on the values ±1, resulting in β k k k Finally, it should be noted that the symbol β k η k β k k 1 k 1 k 1 k 1. η k β k 1 is defined by the inductive hyothesis, comleting the roof of the theorem. k 1

11 Generalization of Scholz s law 593 Theorem 3.1 holds regardless of the choice of rime above. Noting that the left-hand side of the law is indeendent of, we see that the residuacity of β t only deends uon the rime. Hence, we may write β t β t t 1 t 1, allowing us to interret our law as a true rational recirocity law. When 1 mod, it is known that η ε h for an odd integer h, and a roof can be found in Proosition 3. of [11]. In articular, if 1, then β η ε, so that Theorem 3.1 results in Scholz s recirocity law. The octic case of Theorem 3.1 states that if 1 mod 8 and 1, β8 then η8 η. 8 8 The searation of the last residue symbol is justified since η is a uadratic residue by Scholz s recirocity law. Our law takes on a simler form than that of Buell and Williams and comaring the two octic laws results in the following corollary. Corollary 3.. If 1 mod 8 are distinct rimes satisfying 1, then η8 ε Nε h/, where O K is any rime above. In conclusion, we note that Lemmermeyer commented at the end of [10] that he had generalized Scholz s recirocity law to all number fields with odd class number in the strict sense. Lemmermeyer s generalization has not been ublished, but has aeared in his online notes on class field towers. His generalization is uite different from ours and does not lend itself to an easy comarison. Desite the differences in the two generalizations, all of the work contained here was motivated by the techniues used by Lemmermeyer [11] leading u to his roof of Scholz s Recirocity Law.

12 59 Mark Budden, Jeremiah Eisenmenger, Jonathan Kish References [1] D. Buell and K. Williams, Is There an Octic Recirocity Law of Scholz Tye?. Amer. Math. Monthly , [] D. Buell and K. Williams, An Octic Recirocity Law of Scholz Tye. Proc. Amer. Math. Soc , [3] D. Estes and G. Pall, Sinor Genera of Binary Quadratic Forms. J. Number Theory , 1 3. [] R. Evans, Residuacity of Primes. Rocky Mountain J. of Math , [5] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory. nd edition, Graduate Texts in Mathematics 8, Sringer-Verlag, [6] G. Janusz, Algebraic Number Fields. nd ed., Graduate Studies in Mathematics 7, American Mathematical Society, Providence, RI, [7] E. Lehmer, On the Quadratic Character of some Quadratic Surds. J. Reine Angew. Math , 8. [8] E. Lehmer, Generalizations of Gauss Lemma. Number Theory and Algebra, Academic Press, New York, 1977, [9] E. Lehmer, Rational Recirocity Laws. Amer. Math. Monthly , [10] F. Lemmermeyer, Rational Quartic Recirocity. Acta Arith , [11] F. Lemmermeyer, Recirocity Laws. Sringer Monograhs in Mathematics, Sringer- Verlag, Berlin, 000. [1] A. Scholz, Über die Lösbarkeit der Gleichung t Du. Math. Z , [13] T. Schönemann, Theorie der Symmetrischen Functionen der Wurzeln einer Gleichung. Allgemeine Sätze über Congruenzen nebst einigen Anwendungen derselben. J. Reine Angew. Math , [1] K. Williams, On Scholz s Recirocity Law. Proc. Amer. Math. Soc. 6 No , 5 6. [15] K. Williams, K. Hardy, and C. Friesen, On the Evaluation of the Legendre Symbol A+B m. Acta Arith , Mark Budden Deartment of Mathematics Armstrong Atlantic State University Abercorn St. Savannah, GA USA Mark.Budden@armstrong.edu URL: htt:// Jeremiah Eisenmenger Deartment of Mathematics University of Florida PO Box Gainesville, FL USA eisenmen@math.ufl.edu Jonathan Kish Deartment of Mathematics University of Colorado at Boulder Boulder, CO USA jonathan.kish@colorado.edu