In this paper, we study the p-rank of the tame kernels of pure cubic fields. In particular, we prove that for a fixed positive integer m, there exist infinitely many pure cubic fields whose 3-rank of the tame kernel e...In this paper, we study the p-rank of the tame kernels of pure cubic fields. In particular, we prove that for a fixed positive integer m, there exist infinitely many pure cubic fields whose 3-rank of the tame kernel equal to m. As an application, we determine the 3-rank of their tame kernels for some special pure cubic fields.展开更多
Let F = Q（x/P）, where p = 8t ＋ 1 is a prime. In this paper, we prove that a speclm case of Qin＇s conjecture on the possible structure of the 2-primary part of K2OF up to 8-rank is a consequence of a conjecture of ...Let F = Q（x/P）, where p = 8t ＋ 1 is a prime. In this paper, we prove that a speclm case of Qin＇s conjecture on the possible structure of the 2-primary part of K2OF up to 8-rank is a consequence of a conjecture of Cohen and Lagarias on the existence of governing fields. We also characterize the 16-rank of K2OF, which is either 0 or 1, in terms of a certain equation between 2-adic Hilbert symbols being satisfied or not.展开更多
In this paper, we discuss a method to compute the tame kernel of a number field. Confining ourselves to an imaginary quadratic field, we prove that _υ:K_2~S F/K_2~S F→k~* is bijective when N_υ>8δ_D^6.
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
Let F be a pure quintic field. In this paper, we present some results for the p-rank of K2OF, where p is an odd prime number. In particular, the 5-rank of K20F is studied by the reflection theorem. Some explicit resul...Let F be a pure quintic field. In this paper, we present some results for the p-rank of K2OF, where p is an odd prime number. In particular, the 5-rank of K20F is studied by the reflection theorem. Some explicit results on the 5-rank of K20F are given in some special cases.展开更多
In this paper,we present some explicit formulas for the 3-rank of the tame kernels of certain pure cubic number fields,and give the density results concerning the 3-rank of the tame kernels.Numerical examples are give...In this paper,we present some explicit formulas for the 3-rank of the tame kernels of certain pure cubic number fields,and give the density results concerning the 3-rank of the tame kernels.Numerical examples are given in Tables 1 and 2.展开更多
Let F be a cubic cyclic field with t(2)ramified primes.For a finite abelian group G,let r3(G)be the 3-rank of G.If 3 does not ramify in F,then it is proved that t-1 r3(K2O F)2t.Furthermore,if t is fixed,for any s sati...Let F be a cubic cyclic field with t(2)ramified primes.For a finite abelian group G,let r3(G)be the 3-rank of G.If 3 does not ramify in F,then it is proved that t-1 r3(K2O F)2t.Furthermore,if t is fixed,for any s satisfying t-1 s 2t-1,there is always a cubic cyclic field F with exactly t ramified primes such that r3(K2O F)=s.It is also proved that the densities for 3-ranks of tame kernels of cyclic cubic number fields satisfy a Cohen-Lenstra type formula d∞,r=3-r2∞k=1(1-3-k)r k=1(1-3-k)2.This suggests that the Cohen-Lenstra conjecture for ideal class groups can be extended to the tame kernels of cyclic cubic number fields.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.10971091 and 10871088)Specialized Research Fund for the Doctoral Program of Higher Education(Grant Nos.200802840003 and 200802841042)
文摘In this paper, we study the p-rank of the tame kernels of pure cubic fields. In particular, we prove that for a fixed positive integer m, there exist infinitely many pure cubic fields whose 3-rank of the tame kernel equal to m. As an application, we determine the 3-rank of their tame kernels for some special pure cubic fields.
基金Supported by NSFC(Grant Nos.11201225,11271177 and 11171141)
文摘Let F = Q（x/P）, where p = 8t ＋ 1 is a prime. In this paper, we prove that a speclm case of Qin＇s conjecture on the possible structure of the 2-primary part of K2OF up to 8-rank is a consequence of a conjecture of Cohen and Lagarias on the existence of governing fields. We also characterize the 16-rank of K2OF, which is either 0 or 1, in terms of a certain equation between 2-adic Hilbert symbols being satisfied or not.
文摘In this paper, we discuss a method to compute the tame kernel of a number field. Confining ourselves to an imaginary quadratic field, we prove that _υ:K_2~S F/K_2~S F→k~* is bijective when N_υ>8δ_D^6.
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This paper was partially supported by National Natural Science Foundation of China under Grant 11301071 and 11471162.
文摘Let F be a pure quintic field. In this paper, we present some results for the p-rank of K2OF, where p is an odd prime number. In particular, the 5-rank of K20F is studied by the reflection theorem. Some explicit results on the 5-rank of K20F are given in some special cases.
基金supported by National Natural Science Foundation of China (Grant No.10871088)Speialized Research Fund for the Doctoral Program of Higher Education (Grant No.200802840003)the Cultivation Fund of the Key Scientific and Technical Innovation Project,Ministry of Education of China(Grant No.708044)
文摘In this paper,we present some explicit formulas for the 3-rank of the tame kernels of certain pure cubic number fields,and give the density results concerning the 3-rank of the tame kernels.Numerical examples are given in Tables 1 and 2.
基金supported by National Natural Science Foundation of China (Grant Nos. 11201225,11271177,10971091 and 11171141)Natural Science Foundation of the Jiangsu Province (Grant Nos. BK2010007 and BK2010362)Program for New Century Excellent Talents in University (Grant No. NCET-100471)
文摘Let F be a cubic cyclic field with t(2)ramified primes.For a finite abelian group G,let r3(G)be the 3-rank of G.If 3 does not ramify in F,then it is proved that t-1 r3(K2O F)2t.Furthermore,if t is fixed,for any s satisfying t-1 s 2t-1,there is always a cubic cyclic field F with exactly t ramified primes such that r3(K2O F)=s.It is also proved that the densities for 3-ranks of tame kernels of cyclic cubic number fields satisfy a Cohen-Lenstra type formula d∞,r=3-r2∞k=1(1-3-k)r k=1(1-3-k)2.This suggests that the Cohen-Lenstra conjecture for ideal class groups can be extended to the tame kernels of cyclic cubic number fields.
