In this paper, we study the real quadratic function fields K=k(D), given a necessary and sufficient condition for the ideal class group H(K) of any real quadratic function field K to have a cyclic subgroup of order n,...In this paper, we study the real quadratic function fields K=k(D), given a necessary and sufficient condition for the ideal class group H(K) of any real quadratic function field K to have a cyclic subgroup of order n, and obtained eight series of such fields. The ideal class numbers h(O K) of K in the series all have a factor n.[展开更多
It is well known that there is a close connection between tame kernels and ideal class groups of number fields. However, the latter is a very difficult subject in number theory. In this paper, we prove some results co...It is well known that there is a close connection between tame kernels and ideal class groups of number fields. However, the latter is a very difficult subject in number theory. In this paper, we prove some results connecting the p^n-rank of the tame kernel of a cyclic cubic field F with the p^n-rank of the coinvariants of μp^n×CI（δE,T） under the action of the Galois group, where E = F（ζp^n ） and T is the finite set of primes of E consisting of the infinite primes and the finite primes dividing p. In particular, if F is a cyclic cubic field with only one ramified prime and p = 3, n = 2, we apply the results of the tame kernels to prove some results of the ideal class groups of E, the maximal real subfield of E and F（ζ3）.展开更多
Imaginary cyclic fields of degree p-1 which have two distinct unramified cyclic extensions of degree p are produced using elementary properties of the Lucas sequences. An infinite family of imaginary cyclic fields of ...Imaginary cyclic fields of degree p-1 which have two distinct unramified cyclic extensions of degree p are produced using elementary properties of the Lucas sequences. An infinite family of imaginary cyclic fields of degree p-1 are then given with the p-rank of the ideal class groups of at least two.展开更多
In this paper, the theory of continued fractions of algebraic functions will be used to give a general theorem on lower bounds for class numbers of real quadratic function fields K=k(D). The bounds are given more expl...In this paper, the theory of continued fractions of algebraic functions will be used to give a general theorem on lower bounds for class numbers of real quadratic function fields K=k(D). The bounds are given more explicitly for six types of real quadratic function fields. As a consequence, six classes of real quadratic function fields with ideal class number greater than one are given.[展开更多
The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the...The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG)2 + cS, then h(D) ≥ degS/degP; if D = (A m + a)2 + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q . In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.展开更多
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
Necessary and sufficient condition on real quadratic algebraic function fields K is given for theirideal class groups H(K) to contain cyclic subgroups of order n. And eight series of such real quadratic functionfields...Necessary and sufficient condition on real quadratic algebraic function fields K is given for theirideal class groups H(K) to contain cyclic subgroups of order n. And eight series of such real quadratic functionfields K are obtained whose ideal class groups contain cyclic subgroups of order n. In particular, the ideal classnumbers of these function fields are divisible by n.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
A necessary condition is presented for the ideal class group of an imaginary quadratic function field K=k(D) (k=F-q(x), 2q) to be of exponent ≤2. The condition is proved to be sufficient in some cases. An analogue of...A necessary condition is presented for the ideal class group of an imaginary quadratic function field K=k(D) (k=F-q(x), 2q) to be of exponent ≤2. The condition is proved to be sufficient in some cases. An analogue of Louboutin’s result in function field case is particularly presented.展开更多
Quadratic-field cryptosystem is a cryptosystem built from discrete logarithm problem in ideal class groups of quadratic fields(CL-DLP). The problem on digital signature scheme based on ideal class groups of quadratic ...Quadratic-field cryptosystem is a cryptosystem built from discrete logarithm problem in ideal class groups of quadratic fields(CL-DLP). The problem on digital signature scheme based on ideal class groups of quadratic fields remained open, because of the difficulty of computing class numbers of quadratic fields. In this paper, according to our researches on quadratic fields, we construct the first digital signature scheme in ideal class groups of quadratic fields, using q as modulus, which denotes the prime divisors of ideal class numbers of quadratic fields. Security of the new signature scheme is based fully on CL-DLP. This paper also investigates realization of the scheme, and proposes the concrete technique. In addition, the technique introduced in the paper can be utilized to realize signature schemes of other kinds.展开更多
文摘In this paper, we study the real quadratic function fields K=k(D), given a necessary and sufficient condition for the ideal class group H(K) of any real quadratic function field K to have a cyclic subgroup of order n, and obtained eight series of such fields. The ideal class numbers h(O K) of K in the series all have a factor n.[
文摘It is well known that there is a close connection between tame kernels and ideal class groups of number fields. However, the latter is a very difficult subject in number theory. In this paper, we prove some results connecting the p^n-rank of the tame kernel of a cyclic cubic field F with the p^n-rank of the coinvariants of μp^n×CI（δE,T） under the action of the Galois group, where E = F（ζp^n ） and T is the finite set of primes of E consisting of the infinite primes and the finite primes dividing p. In particular, if F is a cyclic cubic field with only one ramified prime and p = 3, n = 2, we apply the results of the tame kernels to prove some results of the ideal class groups of E, the maximal real subfield of E and F（ζ3）.
基金Supported by the Japan Society for the Promotion of Science (JSPS) (No. 14540030) the JSPS Research Fellowships for Young Scientists
文摘Imaginary cyclic fields of degree p-1 which have two distinct unramified cyclic extensions of degree p are produced using elementary properties of the Lucas sequences. An infinite family of imaginary cyclic fields of degree p-1 are then given with the p-rank of the ideal class groups of at least two.
文摘In this paper, the theory of continued fractions of algebraic functions will be used to give a general theorem on lower bounds for class numbers of real quadratic function fields K=k(D). The bounds are given more explicitly for six types of real quadratic function fields. As a consequence, six classes of real quadratic function fields with ideal class number greater than one are given.[
文摘The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG)2 + cS, then h(D) ≥ degS/degP; if D = (A m + a)2 + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q . In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.
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This work was supported by the National Natural Science Foundation of China(Grant No.10071041).
文摘Necessary and sufficient condition on real quadratic algebraic function fields K is given for theirideal class groups H(K) to contain cyclic subgroups of order n. And eight series of such real quadratic functionfields K are obtained whose ideal class groups contain cyclic subgroups of order n. In particular, the ideal classnumbers of these function fields are divisible by n.
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A necessary condition is presented for the ideal class group of an imaginary quadratic function field K=k(D) (k=F-q(x), 2q) to be of exponent ≤2. The condition is proved to be sufficient in some cases. An analogue of Louboutin’s result in function field case is particularly presented.
文摘Quadratic-field cryptosystem is a cryptosystem built from discrete logarithm problem in ideal class groups of quadratic fields(CL-DLP). The problem on digital signature scheme based on ideal class groups of quadratic fields remained open, because of the difficulty of computing class numbers of quadratic fields. In this paper, according to our researches on quadratic fields, we construct the first digital signature scheme in ideal class groups of quadratic fields, using q as modulus, which denotes the prime divisors of ideal class numbers of quadratic fields. Security of the new signature scheme is based fully on CL-DLP. This paper also investigates realization of the scheme, and proposes the concrete technique. In addition, the technique introduced in the paper can be utilized to realize signature schemes of other kinds.
