AMBIGUOUS CLASSES IN QUADRATIC FIELDS


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1 MATHEMATICS OF COMPUTATION VOLUME, NUMBER 0 JULY 99, PAGES 0 AMBIGUOUS CLASSES IN QUADRATIC FIELDS R. A. MOLLIN Dedicated to the memory ofd. H. Lehmer Abstract. We provide sufficient conditions for the class group of a quadratic field (with positive or negative discriminant) to be generated by ambiguous ideals. This investigation was motivated by a recent result of F. HalterKoch, which we show is false.. Introduction The principal result is the provision of sufficient conditions for the class group of quadratic fields to be generated by certain prescribed ambiguous ideals in terms of certain canonical quadratic polynomials. We were motivated by the principal result of [], which contains a serious error. We therefore provide a counterexample to Theorem. of [], which motivates the discussion. Although the aforementioned result is false, we provide a list (which we conjecture to be complete) of all the polynomials which satisfy the prime producing hypothesis (but not necessarily the conclusion) of []. We maintain that this is of interest in its own right.. Notation and preliminaries Let Q(Vd) be a quadratic field, where d is a (positive or negative) squarefree integer, and let A = 4d/a be its discriminant, where a = if d = (mod 4) and a = otherwise. Thus the radicand d is the squarefree kernel of A. Let [a, ß] = az ßZ ; then the maximal order cfa of Q(Vd) is [, coa], where coa = (a  + \fd)/a. It is well known that / is a nonzero ideal in cfa if and only if I = [a, b + ccoa], where a, b, c e Z with c\b, c\a and ac\n(b+ccoa), where N is the norm, i.e., N(a) = aä, where a. is the algebraic conjugate of a. The ideal / is called primitive if c = and a > 0. In this case, a is the smallest positive integer in / and a = N(I) = (rfa: I). Let CA denote the class group of cfa. Equivalence in CA is denoted by / ~ J (by which we mean that there are nonzero elements ai and a of cfa with axi = aj). We denote the order of CA by ha, the class number of cfa. Principal ideals (generated by a single element a) are denoted by (a). The following is well known. Received by the editor May 9, 99 and, in revised form, September 0, Mathematics Subject Classification. Primary R, R9, Y40. Key words and phrases. Ambiguous ideal, quadratic field, class number, exponent. 99 American Mathematical Society 008/9 $.00 + $. per page
2 R. A. MOLLIN Theorem.. Every class of CA contains a primitive ideal I with N(I) < MA, where MA = < \ ypm ifa<0,,_ \y/äß if A>0. Based upon the above, the following improves upon Theorem. of []. Theorem.. The class group CA is generated by the primitive prime ideals a with N(>) < MA. Proof. By Theorem. there is an ideal / in each class with N(I) < MA. Each such ideal is divisible by a primitive prime ideal, and N(9 ) < N(I) < MA. The result now follows. D Throughout the paper, if q is a positive squarefree divisor of A, then f? denotes the unique cfaprime ideal above q, or simply if no confusion with other divisors of A arises.. Prime quadratics As mentioned above, we begin with a counterexample to Theorem. of [], which motivates the discussion leading to criteria for CA to be generated by ambiguous ideals (given in terms of certain local primality conditions for the polynomials which we now define). Definition.. Let q be a positive squarefree divisor of a discriminant A; then fn,,w = ax + (a \)qx + ((« )<  A), where a = if 4q divides A and a = otherwise. Also, set AA = li(mal)]. Example.. The claim of [, Theorem., p. ] is that if, under the assumption of Definition., we have that \FA<q(x)\ is or prime for all integers x with 0 < x < AA, then Ck = {, <S} The following example contradicts this assertion. Let A = 8 = 9 =  4 and q = ; then FA_q(x) = x+ *. Here, [MA\ = and AA =, so \FA,q(x)\ =, 9, 89 and 9 for x = 0,,, and, respectively. However, CA ^ {, ^}. In fact f) ~ and CA = (é'i) = (áf), where (x) denotes the cyclic group generated by x. Remark.. For the interested reader the error in the proof of [, Theorem., p. ] lies in the assumption that (in HalterKoch's notation) J^ is primitive. (For instance, in Example., fq = &xi and Ss = ß; whence, J^f' = f^ds is definitely not primitive.) We have compiled in Table. a list of all positive radicands for which \FA,q(x)\ is or prime for all integers x with 0 < x < AA. Therein, the radicands 08, 9 and 08 are also counterexamples to [, Theorem.]. Now we prove a result which gives sufficient conditions for CA to be generated by ambiguous ideals. First we need a preliminary result which generalizes [, Lemma., p. ].
3 ambiguous classes in quadratic fields Lemma.. Let q > be a squarefree divisor of a discriminant A. If p is a prime, then FAq(x) = 0 (mod p) for some integer x > 0 if and only if (A/p) /  and p does not divide q. Proof. If (A/p) t  and p = does not divide q, then choose x = 0 if (q(a  )  A)4q is even, and x = otherwise. Now assume p >. Thus, there is an integer y such that A/q = qy (mod p). By replacing y by p  y if necessary, we may assume that y = a  (mod ). Setting y = x + a , we get that A/q = q(x + a  I) (mod p), i.e., ., Fa,<i(x) = qxl + qx(a l) + j((a l)q  A) = 0 (mod p). Conversely, if FA>q(x) = 0 (mod p), then A = [q(x + a  l)] (mod p), whence (A/p) # . Moreover, if p divides q, then p divides A, whence p =. Therefore, A = 0 (mod 4). If A = 8 (mod ), then qx  A/4q is odd for all x > 0, a contradiction. Therefore, A = (mod ), in which case (q A)/4q must be even (since a = and q is even in this case). Therefore, q = A (mod ), whence (q/) = (mod 4), which is absurd. D Theorem.. Let q > for < i < n be pairwise relatively prime, squarefree divisors of a discriminant A. If, for each prime p < MA with (A/p) ^  and p ^ q, for any positive i < n there exists a q = Ylie,? Qi for some S? ç {,,...,«} such that \FA<q(x)\  p for some nonnegative integer x, then CA = n}. Proof. By Theorem., CA is generated by the primitive prime ideals P with N( P) = p < MA and (A/p) / . If p / q for any positive i < n, then by hypothesis, IFa.^MI =P for some integer x > 0 (with q as above). Therefore, \(q(x + a I) A/q) =p. Thus, \[(q(x + a  l))a]=pq. Now set {qx if a =, qx + (ql)/ if a and q is odd, q(x+l)/ if a = and q is even. Hence, M  [pq, b + coa] is primitive with N(b + coa) = pq, i.e., ^f ~ (since, in fact, W (b + coa)), whence P and the result follows. D Remark.. The case where n = in Theorem. provides a correct version of Theorem. of []. We note that the radicand 8 in Example. does not allow us to choose q  because neither \FAX(x)\ nor FA,q(x)\ yields the value or for any nonnegative integer x < AA =. However, if we choose q = or q =, then the hypothesis of Theorem. is satisfied, since IFa,() = and Fa,() =, with the observation that and are the only noninert primes less than MA. Therefore, we arrive at the correct conclusion, i.e., C8 = m = (S,). Remark.. Theorem. of [] becomes correct under the assumption q = or q prime, < >0, Q \ f Now we give a sequence of examples to illustrate Theorem.. Example.. Let A = = 89 = 4 + and set q =. Since [MA\ = and the only noninert prime p < MA with p ^ is p , then
4 8 R. A. MOLLIN the fact that FA xi(l) = implies that the hypothesis of Theorem. holds. Thus, CA = i<sù). Example.. Let A = 4 9 = 4 ; then [MA\ = and AA =. Let <?i = and # =. Since FA^X(l) =, which is the only prime p < MA, p j= with p noninert, then by Theorem. ca = m x i&). Example.4. Let A = 4 = 4 ; then [MA\ = 0 and AA = 4. If qx =, q  and qi =, then the only noninert primes p < MA with p t q for i =,, are,, and 9. Since FA,i(l) =, FAX0(0) =, ^a,is()= and Fa,() = 9, then by Theorem. CA = ( f> x (Ss) x (<f). The following consequences of Theorem. show its high degree of applicability to results in the literature. Corollary. [, Theorem, p. ]. If A = 4(4l ± ) > 8 and FA>(.x) = \x  d/\ is or prime for all integers x with 0 < x < \fd/, then ha\. Proof. By Theorem. and Lemma., CA = {, (S). ~, since V((/ ± + VK)/)\ =. D Corollary. [, Theorem, p. ]. If A = 4((l + l) ± ) with I > 0 and IFa^MI = x + x + (  A)/ is or prime for all integers x with 0<x< (VÄ+ l)/, then ha=l. Proof. By Theorem. and Lemma., CA However, <S ~, since \N(l+ +^) =. D The following is [, Conjecture, p. 8]. Corollary.. Let A = I  q = (mod 8), where q is an odd prime dividing I U \Fis,q(x)\ = \qx + qx + (q  A)/4q\ is or prime for all integers x with 0 < jc < y/d  /4  \, then ha =. Proof. By Theorem. and Lemma., CA = {, Sq}. ~, since N(l + \/Ä) = q. D The following is [, Conjecture 4, p. 9]. Corollary.4. Let A I ± 4q, where q is an odd prime dividing I. If \qx + qx + (q  A)/4<? is or prime for all integers x with 0 < x < (>/A~=T+l)/, then ha=\. Proof. Since \N((l + \/Ä)/)\ = q, then the result follows as above. Remark.4. The types of forms in Corollaries..4 are called extended RichaudDegert (ERD)types, i.e., those of the form d = A/< = I + r, where r divides 4/. In [, 4] we found all such A's with ha = under the assumption of a suitable Riemann hypothesis, and in [] we were able to eliminate the Riemann hypothesis and show that our list is complete (with one possible exceptional value of A, whose existence would be a counterexample to that Riemann hypothesis). The following answers a question about ERD types posed in [, Remark, p. ]. D
5 AMBIGUOUS CLASSES IN QUADRATIC FIELDS 9 Table.. This table lists all radicands 0 < d < 0 such that Fa,9(x) is or prime for all nonnegative integers x < AA, where q is a positive squarefree divisor of the discriminant A. G Example.. Let A = 9 = ; then [MA\ =, AA =. The only noninert prime p <, p ^, is p = and FA () , so by Theorem., CA = (Si). We note that 9 is not an ERDtype. The only other nonerd type appearing in Table. is A = 4. (Note that since Fr4), n(0) =, the only noninert prime p < M^x (with p ^ ), then C4 = (<SXX)). Although the principal result of [] has been shown to be false herein, it is still of interest in its own right to determine all those positive A's and q's such that FAq(x) is or prime for 0 < x < AA. Some serious computational evidence leads us to pose: Conjecture.. Table. is complete, i.e., if d > 44, then \FA,q(x)\ is composite for some nonnegative x < AA. Remark.. It is interesting to note that the only ERD types d, with class number, that do not appear in Table. are d,, 4, 8, 0,, 9, and. Moreover, the only nonerd types which are in Table. are 4 and 9. Also, the values which have class number bigger than are 0,,, 0,, 4,, 8, 8, 0, 0,, 8,, 8,, 0, 8, 8, 8,,,, 40, 4, 8, 9, 9, 08, 4, 8, 9, 08,, 9 and 44, all of which have class number. We note that if FA,q(x) is or prime for all integers x with 0 < x < AA, we cannot even guarantee that ha <. Given Remark.4 and Conjecture. above, we must turn to negative discriminants.
6 0 R. A. MOLLIN Example.. Let A = 0 =  and set q = 0, whence \_MA\ =, and so AA =. Also, Fa,,(jc) = 0x + =,,, 0,, and for x = 0,,,, 4, and, respectively. However, ha = 4. In fact, then CA ^ {, So}. If in Theorem. we take qx = and <? =, then FA(0) =, which is the only noninert prime less than MA (other than and ), so the hypothesis of Theorem. is satisfied and we have that CA = (if) x ( f). We observe that both FA (0) and FA (0) are composite. This illustrates that the generation of Ca by ambiguous ideals dividing q, in general, has less to do with the primeproducing capacity of FA>q(x) for an initial string of x values, as asserted in [], than it does with its local capacity to "hit" certain primes. We have made significant progress with negative discriminants and quadratic polynomials, which will be published at a later date. Acknowledgment The author's research is supported by NSERC Canada grant #A8484. Moreover, the author welcomes the opportunity to thank Mike Jacobson, a student of Hugh Williams, for doing the computations which led to Table.. Bibliography. F. HalterKoch, Primeproducing quadratic polynomials and class numbers of quadratic orders, Computational Number Theory (A. Pethö, M. Pohst, H. C. Williams, and H. Zimmer, eds.), de Gruyter, Berlin, 99, pp S. Louboutin, R. A. Mollin, and H. C. Williams, Class numbers of real quadratic fields, continued fractions, reduced ideals, primeproducing quadratic polynomials and quadratic residue covers, Cañad. J. Math. 44 (99), 9.. R. A. Mollin and H. C. Williams, Prime producing quadratic polynomials and real quadratic fields of class number one, Number Theory (J M. De Koninck and C Levesque, eds.), de Gruyter, Berlin, 989, pp _, On prime valued polynomials and class numbers of real quadratic fields, Nagoya Math. J. (988), 4.. _, Solution of the class number one problem for real quadratic fields of extended Richaud Degert type (with one possible exception), Number Theory (R. A. Mollin, ed.), de Gruyter, Berlin, 990, pp Department of Mathematics, University of Calgary, Calgary, Alberta, Canada TN N4 address :
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