A necessary and sufficient condition is given for the ideal class group H( m } of a real quadratic field Q (m)to contain a cyclic subgroup of order n.Some criteria satisfying the condition are also obtained.And eight ...A necessary and sufficient condition is given for the ideal class group H( m } of a real quadratic field Q (m)to contain a cyclic subgroup of order n.Some criteria satisfying the condition are also obtained.And eight types of such fields are proved to have this property,e.g.fields with m=(zn+t+12)+4t(with t|zn-1),which contains the well-known fields with m=4zn+1 and m=z2n+4 as special cases.展开更多
LetK 6 be a real cyclic sextic number field, andK 2,K 3 its quadratic and cubic subfield. Leth(L) denote the ideal class number of fieldL. Seven congruences forh - =h (K 6)/(h(K 2)h(K 3)) are obtained. In particular, ...LetK 6 be a real cyclic sextic number field, andK 2,K 3 its quadratic and cubic subfield. Leth(L) denote the ideal class number of fieldL. Seven congruences forh - =h (K 6)/(h(K 2)h(K 3)) are obtained. In particular, when the conductorf 6 ofK 6 is a primep, $Ch^ - \equiv B\tfrac{{p - 1}}{6}B\tfrac{{5(p - 1)}}{6}(\bmod p)$ , whereC is an explicitly given constant, andB n is the Bernoulli number. These results on real cyclic sextic fields are an extension of the results on quadratic and cyclic quartic fields.展开更多
In this paper we give a formula for the number of representations of some square-free integers by certain ternary quadratic forms and estimate the lower bound of the 2-power appearing in this number.
Let K be a cyclic quartic number field, and k its quadratic subfield. Let h(L) denote theideal class number of field L. Ten congruenees for h^- = h(K)/h(k) are obtained. In par-ticular, if K = Q((p+s(p^(1/2))))^(1/2) ...Let K be a cyclic quartic number field, and k its quadratic subfield. Let h(L) denote theideal class number of field L. Ten congruenees for h^- = h(K)/h(k) are obtained. In par-ticular, if K = Q((p+s(p^(1/2))))^(1/2) with the prime number p = r^2+s^2 and s is even, then C_1h^-≡B_((p-1)/_4)B_(3(p-1)/4) (mod p) for p≡1 (mod 8); and C_2h^-≡E_((p-5)/8)E_((3p-7)/8)(mod p) for p≡5 (mod 8)where B_n and E_n are the Bernoulli and the Euler numbers. If the real K = Q((v(5+2(5^(1/2))))^(1/2),then C_3h^-≡h(Q((-v)^(1/2))) h (Q((-5v)^(1/2))) (mod 5). If 3 ramifies in K = Q(θ^(1/2)), then C_4h(K)≡h(K~*) (mod 3) with K~* = Q((-3θ^(1/2))). All the above C_i are explicitly given constants.Some relations between the factors of class numbers h^- are also obtained. These results forcyclic quartic fields are an extension of the results for quadratic fields obtained by Ankeny-Artin-Chowla, Kiselev, Carlitz and Lu Hong-wen from 1948 to 1983.展开更多
THE famous Cohen-Lenstra heuristics aroused wide insterest and research. Here for a certaintype of real quadratic fields with elements P of potential order p in their ideal classes, modifi-cations of the Cohen-Lenstra...THE famous Cohen-Lenstra heuristics aroused wide insterest and research. Here for a certaintype of real quadratic fields with elements P of potential order p in their ideal classes, modifi-cations of the Cohen-Lenstra heuristics for the probability that the class number h is a multipleof p, and the probability that P is of order p, are presented. Via a quite large amount ofcomputations, it was found that both of these probability predictions agree fairly well with thenumerical data.展开更多
Series of results about ideal class groups H （m） and class numbers h （m） of real quadratic fields Q (m<sup>1/2</sup>) can be obtained from [6]. Some of them will be shown in this note. We denote by C&l...Series of results about ideal class groups H （m） and class numbers h （m） of real quadratic fields Q (m<sup>1/2</sup>) can be obtained from [6]. Some of them will be shown in this note. We denote by C<sub>n</sub> =Z/nZ the cyclic group of order n. Let m ∈ Z denote a square free positive integer, and let z<sub>1</sub>, z, t ∈Z be arbitrary integers with z<sub>1</sub> odd and t】0.展开更多
Let the chromatic number of G, the edge chromatic number of G and thetotal chromatic number of G be denoted by x（G）, x<sub>1</sub>（G） and x<sub>2</sub>（G）, respectively. Forany simple gra...Let the chromatic number of G, the edge chromatic number of G and thetotal chromatic number of G be denoted by x（G）, x<sub>1</sub>（G） and x<sub>2</sub>（G）, respectively. Forany simple graph G of order p and its complement G, the following inequalities of theNordhaus-Gaddum class are obtained:（i）|2p<sup>1/2</sup>|-ε<sub>1</sub>≤x（G）+x<sub>1</sub>（G）≤2p-2 and 0≤x（G）·x<sub>1</sub>（G）≤（p-1）<sup>2</sup> for p≥2,（ii）|2p<sup>1/2</sup>|+ε<sub>1</sub>≤x（G）+x<sub>2</sub>（G）≤2p-1 and 0≤x（G）·x<sub>2</sub>（G）≤p（p-1） for p≥3,（iii）p≤x<sub>1</sub>（G）+x<sub>2</sub>（G）≤2p-1 and 0≤x<sub>1</sub>（G）·x<sub>2</sub>（G）≤p（p-1） for p≥3,where ε<sub>1</sub>=0, if p<sup>1/2</sup> is an odd integer, 1, otherwise,ε<sub>2</sub>=1, if p<sup>1/2</sup> is an even integer, 0, otherwise,and [x] denotes the ceiling of x. We also show that these bounds are sharp for everypositive integer p.展开更多
For real quadratic fields K, especially for fields K of ERD-type, a series of criteria of ideal class numbers h(K)=1 and h(K)】1 will be given via results of Diophantine equations in [1] and continued fraction theory....For real quadratic fields K, especially for fields K of ERD-type, a series of criteria of ideal class numbers h(K)=1 and h(K)】1 will be given via results of Diophantine equations in [1] and continued fraction theory. The problem of class numbers of real quadratic fields, after Gauss’conjecture, has been studied. For example, Lu Hong-wen展开更多
基金Project supported by the National Natural Science Foundation of China.
文摘A necessary and sufficient condition is given for the ideal class group H( m } of a real quadratic field Q (m)to contain a cyclic subgroup of order n.Some criteria satisfying the condition are also obtained.And eight types of such fields are proved to have this property,e.g.fields with m=(zn+t+12)+4t(with t|zn-1),which contains the well-known fields with m=4zn+1 and m=z2n+4 as special cases.
基金Project supported by the National Natural Science Foundation of China (Grant No. 19771052).
文摘LetK 6 be a real cyclic sextic number field, andK 2,K 3 its quadratic and cubic subfield. Leth(L) denote the ideal class number of fieldL. Seven congruences forh - =h (K 6)/(h(K 2)h(K 3)) are obtained. In particular, when the conductorf 6 ofK 6 is a primep, $Ch^ - \equiv B\tfrac{{p - 1}}{6}B\tfrac{{5(p - 1)}}{6}(\bmod p)$ , whereC is an explicitly given constant, andB n is the Bernoulli number. These results on real cyclic sextic fields are an extension of the results on quadratic and cyclic quartic fields.
文摘In this paper we give a formula for the number of representations of some square-free integers by certain ternary quadratic forms and estimate the lower bound of the 2-power appearing in this number.
基金Project supported by the National Natural Science Foundation of China.
文摘Let K be a cyclic quartic number field, and k its quadratic subfield. Let h(L) denote theideal class number of field L. Ten congruenees for h^- = h(K)/h(k) are obtained. In par-ticular, if K = Q((p+s(p^(1/2))))^(1/2) with the prime number p = r^2+s^2 and s is even, then C_1h^-≡B_((p-1)/_4)B_(3(p-1)/4) (mod p) for p≡1 (mod 8); and C_2h^-≡E_((p-5)/8)E_((3p-7)/8)(mod p) for p≡5 (mod 8)where B_n and E_n are the Bernoulli and the Euler numbers. If the real K = Q((v(5+2(5^(1/2))))^(1/2),then C_3h^-≡h(Q((-v)^(1/2))) h (Q((-5v)^(1/2))) (mod 5). If 3 ramifies in K = Q(θ^(1/2)), then C_4h(K)≡h(K~*) (mod 3) with K~* = Q((-3θ^(1/2))). All the above C_i are explicitly given constants.Some relations between the factors of class numbers h^- are also obtained. These results forcyclic quartic fields are an extension of the results for quadratic fields obtained by Ankeny-Artin-Chowla, Kiselev, Carlitz and Lu Hong-wen from 1948 to 1983.
文摘THE famous Cohen-Lenstra heuristics aroused wide insterest and research. Here for a certaintype of real quadratic fields with elements P of potential order p in their ideal classes, modifi-cations of the Cohen-Lenstra heuristics for the probability that the class number h is a multipleof p, and the probability that P is of order p, are presented. Via a quite large amount ofcomputations, it was found that both of these probability predictions agree fairly well with thenumerical data.
文摘Series of results about ideal class groups H （m） and class numbers h （m） of real quadratic fields Q (m<sup>1/2</sup>) can be obtained from [6]. Some of them will be shown in this note. We denote by C<sub>n</sub> =Z/nZ the cyclic group of order n. Let m ∈ Z denote a square free positive integer, and let z<sub>1</sub>, z, t ∈Z be arbitrary integers with z<sub>1</sub> odd and t】0.
文摘Let the chromatic number of G, the edge chromatic number of G and thetotal chromatic number of G be denoted by x（G）, x<sub>1</sub>（G） and x<sub>2</sub>（G）, respectively. Forany simple graph G of order p and its complement G, the following inequalities of theNordhaus-Gaddum class are obtained:（i）|2p<sup>1/2</sup>|-ε<sub>1</sub>≤x（G）+x<sub>1</sub>（G）≤2p-2 and 0≤x（G）·x<sub>1</sub>（G）≤（p-1）<sup>2</sup> for p≥2,（ii）|2p<sup>1/2</sup>|+ε<sub>1</sub>≤x（G）+x<sub>2</sub>（G）≤2p-1 and 0≤x（G）·x<sub>2</sub>（G）≤p（p-1） for p≥3,（iii）p≤x<sub>1</sub>（G）+x<sub>2</sub>（G）≤2p-1 and 0≤x<sub>1</sub>（G）·x<sub>2</sub>（G）≤p（p-1） for p≥3,where ε<sub>1</sub>=0, if p<sup>1/2</sup> is an odd integer, 1, otherwise,ε<sub>2</sub>=1, if p<sup>1/2</sup> is an even integer, 0, otherwise,and [x] denotes the ceiling of x. We also show that these bounds are sharp for everypositive integer p.
基金Project supported partially by the National Natural Science Foundation of China.
文摘For real quadratic fields K, especially for fields K of ERD-type, a series of criteria of ideal class numbers h(K)=1 and h(K)】1 will be given via results of Diophantine equations in [1] and continued fraction theory. The problem of class numbers of real quadratic fields, after Gauss’conjecture, has been studied. For example, Lu Hong-wen
