Séminaire Delange-Pisot-Poitou. Théorie des nombres H. W., J R . L ENSTRA Artin’s conjecture on primes with prescribed primitive roots Séminaire Delange-Pisot-Poitou. Théorie des nombres, tome exp. no 14, p. 1-8 18, no 1 (1976-1977), <http://www.numdam.org/item?id=SDPP_1976-1977__18_1_A11_0> © Séminaire Delange-Pisot-Poitou. Théorie des nombres (Secrétariat mathématique, Paris), 1976-1977, tous droits réservés. L’accès aux archives de la collection « Séminaire Delange-Pisot-Poitou. Théorie des nombres » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Seminaire DELANGE-PISOT-POITOU (Theorie 1976/77, année, 18e nombres) des n° 14, 14-01 8 p. 31 janvier 1977 ARTIN’S CONJECTURE ON PRIMES WITH PRESCRIBED PRIMITIVE ROOTS H. W. by LENSTRA, Jr 1. Introduc.tion. In for which imply given a square ; infinitely t is density prime numbers, provided that many such proviso which is clearly necessary. It is a of the set of primes primitive root. His conjecture would a takes the place of this clearly necessary Artin’s conjecture. To give about the conjecture a integer non-zero that there exist t ~ and E. ARTIN formula ted 1927, in certain proviso our aim to generalized t ~ see -1 what forms of example, suppose that, besides requiring that t is a primitive root modulo the prime q , we also require that q is in a given arithmetic progression: an q ~ b Under which conditions infinitely many dependonto if, such (b , c mod c on q ? (b , c ~ - 1. ~ . integers, is it reasonable to t , b , c expect that there are We should not think of both requirements on q being infor example, 1 mod 8 , then 2 will be a q is required to be square mod q , hence no primitive root. Similarly, but more subtly, one observes that there are no;primes q which are 3 mod 4 for which 27 is a primitive root : since 27 is a third power, such and 27 would be a square q would have to be mod q . The question The generalized form of Artin’s conjecture we 2 mod 3 , so 11 mod 12 , is to list all such obstructions. have in mind is stated in section 2. The derivation follows Artin’s original heuristics. The status of the conjecture is discussed in section 3. The question under which conditions the set of primes under consideration may be expected to be infinite is answered in section 4. In section 5, we discuss some examples. Among these is an application to the existence of a euclidean algorithm in rings of arithmetic 2. A generalization of Artin’s , Let K be divisor of Next, ly, , and let global field. By a be for all but we are a prime of K group subgroup P p the natural map case of Artin’s subgroup generated- by t . prime K# K - 0 . of K * =P K - {0} . Clear- infinite, finitely generated finitely many p , there is a natural is denoted an non-archimedean we mean a K , interested in those in the p by (2.1) Thus, conjecture. K ; its residue class field W type. and = homomorphism for which W ~ K*p is original conjecture surjective. one should take W = t > , the Let F be (J ~ 8 ~o = This is a non-empty F/K . fied in there is prime of % , subset C ~ G Let a q-integral for all mod q we symbol : denote the Frobenius (p , F/K) (p, F/K) , K, with group 8 . By finite Galois extension of a be F) . x E and it is conjugacy class if a subset which is a p, such that extending F of q union of a p is unrami- conjugacy classes. The condition (p , F/K) C (2.2) C ,p and F can, for suitable choices of C be used to express various requirements polynomials should, or should not, have a zero mod p ;9 that p belongs to a certain arithmetic progression or, if say, K = Q. ~ is represented by a certain binary quadratic form ; or any finite logical combination of such condip : that certain on tions. We denote H by (2.l) conditions "density" density in the Here F, C) = (2.2). and means We are interested in of K formula for the char(K) , Then the formula is classical. For a by n, not divisible primitive n-th root of unity. nitely many primes p of K, satisfying both density of M . let Ln= a positive square- K( , w ~ , (2.3) , Mm ~ M if n n is prime n elements from the "limit" multiple a n M . a that, for all but (2.l) if, and only if, standard argument shows a satisfies p prime nu.mbers ~ ~ char(K). Let ~~in be the set of primes for all prime numbers ~ dividing n, and (2.2). for all Then a p case. The heuristic derivation of such free integer primes in the function field case, and natural density Dirichlet number field the set of m, and of The density E C , and of p fi- satisfying by only finitely many is easily calculated using dif fers M M Tchebotarev’s theorem :1 If Cn bers ~ then, fa = E Gal ( F. L n ~K ~ : dividing if p idL~ n~ 9 is outside So Tchebotarev’s theorem some finite set, implies that M p E prime num- C. if, and only if, d(M ) density the for all exists and of equals a . where we have an if n is squarefree positive integers then the sequence Since i~~ ~an~ has a multiple of m, not divisible by a limit, is the limit of the sets which ~i n , we so if we char(K) , call let n range over all ordered by divisibility, a : this leads to the following conjecture. (2.4) d(M) density CONJECTURE. - The exists and equals a . 3. The status of the conjecture. In 1967, density) conjectural the hypotheses [4-]. tus of the (3.1) the the on sequence of generalized Riemann a conjecture. (-function field, (2.4) then field, then (2.4) is true if f or every generalized Riemann hypothesis. If K number a satisfies the L of is K If hypothesis modulo the Riemann for proved more is a conjecture fields, which was later shown actually proved (~1~, p. 485), it is not general statement in We turn to the number field version of Artin’s a finite curves over WEIL to be correct. From what BILHARZ hard to derive the n is true. In the function field case, BILHARZ by of assumption corrected formula for a Not surprisingly, the various generalizations do not affect the sta- THEOREM. - function (with proved Artin’s original conjecture HOOLEY (2.4) (Cf. ~4~) . Generalizing Hooley’s approach, COOKE and case. implying (2.4) for F/K abelian, modulo certain generalized Riemann hypotheses. Using a device employed in the proof of Tchebotarev’s theorem as given p. 169, one easily reduces (2.4) to the case F/K is abelian, and this gives the statement in (3.1)9 with a different set of Riemann hy- [2] proved WEINBERGER a result (3.2) To If (2.4) is true in the see this, notice that the for the upper d+(M) d+(M) density d(Mn) = a = K , it generally true. is arguments in section M :e Since of for all n F case 1 at least an upper bound (finite set) , we have yield in the limit so n, (2.2) that the condition potheses. A simpler approach is based on the observation may be disregarded in the proof of (2.4), i. e. :o d+(M) a . Applying where at v , F , ~ - C) , we find d+ conjectural density of M’ . Since the case t this to the set denotes the asserts that d(F1 u M’) = a + = r,’I(K , att we so d(M) = a , which proves sult of COOKE and WEINBERGER, 4. The non-vanishing In the (3.2). applications of = K of (2.4) (3.2) a’ ) with a a’ = a , special part of case of the re- (3.1 ). a . of Artin’ s to know under which conditions question, Combining + obtain the number field we F a’ , conclude that d+ ( i~2’ ) >~. ( a I u (M’ ) at least in principle. conjecture, a it is of obvious vanishes. The practical importance following theorem answers this ~q.. Z ) if a a if, and only if, 0 = for some n. Moreover, - is finite. M of the limit = is finite. Thus it suffices to prove that some 0 an 0 implies a 0 is obvious from the monotonicity n a . n fiiurther, if a = 0 , then from section 1, we know hence a = 0 that implies an = 0 for n. In the number field case, this is done expressing bers l , a = n - = lim = have ~ then the set 0 = That a THEOREM. - infinite an as a one of the be the n product conjecture, by of those following conditions prime num- o 2 ;9 = divides the discriminant of £ Let product. at least satisfy which in Artints original as the natural map there is ~ prime a K*/K*l is not ord P~ w~ ~ It is easily seen number divisible that the field Ll22 , ... , W rank of that over t K , such that 0 for w E some m n = ~~ ~2 ... it be any prime numbers. Then disjoint composite of the fields ~~ ~ ~2 , . , , ~ ,~t is the linear F.L injective ;9 is finite. Let n n, with b~T Q ; over lying over £ K of p F 03C6(li).lri [Ll. : K] = and that i 1 modulo its torsion subgroup and ; i where r squarefree one checks the denotes the Euler-function. An easy calcu- lation then yields in the limit so the product ranging over all absolutely convergent and only vanish if an does. is (4.1) This proves case cyclotomic extensions finite set of prime numbers and L., are Instead we In a n non-zero, are are constant field can find prime not linearly disjoint over K. rather more can be Since the conclude that product a can argument collapsese of K one we n . case. extensions, numbers ~ and ~;’ and outside any such that L~ delicate treatment is do not wish to enter. Suffice it to say number in a all factors this not dividing numbers £ in the number field In the function field Here prime explicitly required, into the details of which that, as in the number field case, the constructed. principle, it is possible, using given situation : all one has to ~4.1~ do is and its proof, to decide whether considering the Galois extension a ~ 0 F.L and investigating whether C not is empty, where n is the value yielded by the proof of (4.l). For this procedure to be practicable, it is desirable that n be small. This is, essentially, what is achieved by the following or n theorem. (4.2) THEOREM. - Let K . which a 7~ Then product of all prime numbers ~ ~ char(K) if, and only if, there is an automorphism a be the h 0 for of such that (oJF) C ; e prime numbers ~ for all £ In many (3l) W ~ K*i, C C the number (4.2) in h Combining is 1. (4.l)~ (4.2) and obtain the following theorem. we (4.3) applications, ~(~) * for which THEOREM. - Suppose there exists . Then the set LL M is finite if the union ranging prime number l ~ no and, modulo the Riemann for which hypotheses, only if, numbers ~ ~ char(K) those prime over char(K) for which The existence of where the n "only (4.l), of n, in (4.2) is now clearly equivalent to the condition denotes the product of all prime numbers £ with if" part a (4.2) of to show that is obvious. To prove the "if" implies with ~ y...~. for every char(K) . prime numbers 7~ 0 , F(L) . So part it suffices, by squarefree multiple To this end one proves the fol- lowing lemma :1 (4.4) the LEMHA. - Let &#x26; be a degree (4.4) the In Thus, and all F(L) . with prime numbers Then dividing . to j~ ~ find that in the chain of we number, ~ ~ char(K) , is divisible this degree are Applying prime and ... ~ t 1 + assuming that ~ ~ ... fields i-th extension a particular, 1 , so no two of the t + 1 fields coincide. automorphism 03C3 step-wise to an automorphism 03C4 of we can F(03B6h).Ll1 ... degree which this degree is extend the Llt F(03B6h).Lm - T~ By definition of 9 this by lii , for 1 , i > such tha.t ~ identity Cm is divisible on means L ~i ~ for i = Z , 2 , 0 ~ as required. ... ,’ t . t . (5.1) THEOREM - Let b , t~O ~ 1 y - 1 . Let a(t) set of for prime numbers t is To prove this where that 03C3b Ivi does not a theorem, we of the one (3.1) apply over ~ . modulo certain and, generalized following conditions is satisfied : (4.2) and ~~t~( Q, t>, ~( ~ ~ , ~a ~ ~ ~ to the set Q(03B6c) It is mapping’ is finite if and, modulo the Riemann hypotheses, only if, have an automorphism satisfying certain requirements ; here is the of automorphism Then the progression root is finite if primitive Riemann hypotheses, only if, which t and let =1 , denote the discriminant of in the arithmetic q (b , c) positive integers, be c then found Q( c duct of those prime numbers t for which is is the pro- h power in Q . A an straight- forward analysis shows that the only obstructions preventing the existence of such automorphism an are the conditions (a), (b) (c), and and (5,1~ follows. "if"-part of (5.1) has a direct proof, using nothing more than quadratic reciprocity. In fact, it turns out that in each of the situations (a), (b) and (c) the set of primes in question either is empty or only contains the prime 2. But our approach has the advantage that one need not know beforehand the list of exceptional situations : they are just the obstructions encounteIt may be remarked that the red during the construction of tructed one and if in all other situations knows that the list is Our next example is taken from of prime divisors of such that * S ~ 2 . R == (x E Then RS is K (modulo the set Dedekind domain with trices (/ x 1), : x 1)’ with primes an hypotheses. trices. If ces Let K be proof number Then any element of S ~ S~ ! or if Rs suffice. Seven suffice if The a is K of this theorem makes a of p Let K which infinite unit group is x ~ RS . theorem. mann cons- S be a finite set prime divisors, Let for all THEOREM. - [2~]. be can hypotheses). of archimedean S~ A theorem of VASERSTEIN asserts that (5.2) the Riemann COOKE and WEINBERGER containing K : a complete o and not in S) . RS . generated by the elementary COOKE and WEINBERGER field, are assume ma- proved the following certain generalized Rie- product of nine elementary maprincipal ideal domain, eight elementary matriis a can be embedded in the field of real numbers. use of Artin’s conjecture with W = RS ’ and em- ploys condition (2.2) conjecture Artin’s W index of the image of Our final a there exist c~ , exists, then map that the Then b is such that for all r , and r is true modulo certain Riemann algorithm generalized Riemann hypotheses euclidean algorithm 0 = RS on is b, i. e., on Rg c ~ c E 0 , y ’~(b~ . or If such a following theorem states hypotheses, and gives, moreover, a [7~]. is S >,. 2 , THEOREM. - Suppose that a algorithm euclidean a ideal domain. The principal a + qc = of the smallest euclidean and that certain of is the existence of with RS a generalization replaced by the condition that the given integer k . Our results easily (0 , 1 , 2 , ...) -~-~ r converse description (5.3) divides Kp situation. in concerns ~0 ~ map : (2.l) ,~r further a general more example Actually abelian. in which needed, is earry over to this F/K with are principal a ideal domain, true. given by the map ~ defined by with n - p n - = 1 if the natural map = 2 else. Moreover, 03B8 weaker statement If the was Let that sume ~(b) . Dividing (b ~ c) - 1 . one can take r one. element r then RS on So it suffices to prove the mod c = b by their greatest c r = ~(c) > 3 0 ;9 if is then using metic progressions ~~~c) ~ is trivial if which R*S ~ (Ru ,/( c ) ) if algorithm common r = 0 for which divisor, we may as- 9 The existence of and an In the number field case, [8]. euclidean a We look for b and C 6 ~. QUEEN that it is the smallest c ~ 0 . b, is on RS . algorithm is du.e to ( 5.3 ) , defined in surjective,y is p obtained by WEINBERGER [7J easily imply statement of ( 5.3 ) . ~(r~ or (5.3) of case results first 2014~ ----~K is the smallest euclidean The function field a ~ a one can 3(c) = then 1 2 : If c is ~(c) a to be r a then prime element surjective, and we can take generalization of Dirichlet’s take 0 c is a of RS unit for unit ; and, finally, theorem on primes in arithr = prime element which is mod c , b which indeed gives ~(r) ~ 2 3 ~(c) . Hence let ment Let R-. , of r r ~(c ) RS = 2 . In this case, it would be sufficient to find with 03B8(r) = 1 and denote the prime divisor of Then the condition ~(r) = 1 r = b K means : a prime ele~ mod c . corresponding to the prime ideal (r) of 14-08 the natural and the condition r = mod b type (2.2), with question is,f whether the of the one the map RS is K sur jective, can, using class field c F a set theory, be translated into K . Thus suitably chosen abelian extension of M F , C) = contains an element out- S . side question This analyzed using is (4.3 ~, for which mod 4 and it turns out that the contains gative : For example, the ring 1 ~ ---~ ( Z[ ~ ~~(r~ ~ prime element r which surjective ; here indeed no * is may be answer ne- is ~(4~ = 2. analysis shows, however, that by a fortunate coincidence the set M can, in our situation, only be finite, modulo Riemann hypotheses, in case we have something better : namely, if b is congruent to a unit mod c , in which case, of The same course, take for we Finally, r this unit. mention that we modulo Riemann hypotheses, our for ’ applied to yield existence theorems, perfect, one-error-aorrccting nrithmctical-codes results may be [3]). REFERENCES (H.). - [1] BILHARZ t. 114, [2] COOKE [3] GOTO [4] HOOLEY (C.). - On Artin’s p. 209-220. [5] LANG [6] QUEEN vorgegebener Primitivwurzel, Math. Annalen, and WEINBERGER (P. J.). - On the construction of division chains in Comm. in Algebra, t. 3, algebraic number fields, with applications to 481-524. 1975, p. (G.) (M.) three, SL2 , and FUKUMURA (T.). - Perfect nonbinary AN codes with distance Inform. and Control, t. 27, 1975, p. 336-348. (S.). - Algebraic pany, 1970. p. [7] [8] 1937, Primdivisoren mit p. 476-492. (C.). - conjecture, number theory. - Reading, Arithmetic euclidean 105-113. J. reine und angew. rings, (P.). - About euclidean rings, J. WEINBERGER (P. J.). - On euclidean rings SAMUEL ber theory", (Proceedings p. of II. W. LENSTRA, Jr 15 Roetersstraat AMSTERDAM C (Pays-Bas) Acta Addison Math., t. 225, 1967, Wesley publishing Com- Arithm., Warszawa, t. 26, 1974, Algebra, t. 19, 1971, p. 282-301. of algebraic integers, "Analytic num- of 321-332. - Providence, American mathematical Society, 1973 Symposia in pure Mathematics, 24). (Texte reçu le 7 février 1977)