Tame Kernel Research Articles

The shortest vector problem and tame kernels of cyclotomic fields

The Shortest Vector Problem and the computation of the tame kernel in algebraic K-theory are two different but very important problems, and up to now no one has found any manipulated relations between them. In this paper, we are successful for the first time to reduce the computation of tame kernel, in particular the computation of elements of the set Cm, the key part in the method proposed by Tate, to the Shortest Vector Problem of some lattice. Then, with the help of PARI library we implement the above idea and develop a program for computing the tame kernels of the cyclotomic fields with class number 1, and as an example, by running the program we obtain the tame kernel of the cyclotomic field Q(ζ7).

On the tame kernels of imaginary cyclic quartic fields with class number one

In this paper, a general architecture has been established for the computation of K2OF, the tame kernel of F, for imaginary cyclic quartic field F=Q(−(D+BD)) with class number one, in particular with large discriminants. As a result, it is proved that K2OF is trivial in the following three cases: B=1, D=2 or B=2, D=13 or B=2, D=29. In the last case, the discriminant of F is 24389. Hence, it can be claimed that the architecture also works for the computation of the tame kernel of a number field with discriminant less than 25000.

Tame kernels of cubic and sextic fields

Let K be a non-Galois cubic field, and let F denote the normal closure of K/Q or a sextic cyclic field. In this paper, we establish some relations between the p-rank of K2OK (resp. K2OF) and the p-rank of the ideal class groups of some subfields of K(ζp) (resp. F(ζp)). In the case of p=3, we obtain estimates for the p-ranks of tame kernels K2OK (resp. K2OF).

On tame kernels and second regulators of number fields and their subfields

The relations between the tame kernels and second regulators of a number field F Galois over Q, in particular the fields of degree 8 over Q, and of some of its subfields are investigated through the action of the Galois groups on the tame kernels. In the case of the number fields with Galois group D2p, a result of Browkin and Gangl is improved.

On the 2-primary part of tame kernels of real quadratic fields

Let \(F = Q\left( {\sqrt p } \right)\), where p = 8t+1 is a prime. In this paper, we prove that a special 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.

The tame kernel of $\mathbb {Q}(\zeta _{5})$ is trivial

The tame kernel of $\mathbb {Q}(\zeta _{5})$ is trivial

On the p-rank of tame kernel of number fields

In this paper, the relations between p-ranks of the tame kernel and the ideal class group for a general number field are investigated. As a result, nearly all of Browkin's results about quadratic fields are generalized to those for general number fields. In particular, a p-rank formula between the tame kernel and the ideal class group for a totally real number field of odd degree is obtained when p is a Fermat prime. As an example, the case of cyclic quartic fields is considered in more details. More precisely, using the results on cyclic quartic fields, we give some results connecting the p-rank(K2OF) with the p-rank(Cl(OE)), where F is a cyclic quartic field and E is an appropriate subfield of F(ζp).

On the splitting of the exact sequence, relating the wild and tame kernels

Let k be a number field. For an odd prime p and an integer i≥2, let Шe´t2(k,Zp(i)) denote the étale wild kernel of k (corresponding to p and i). Then Шe´t2(k,Zp(i)) is contained in the finite group He´t2(ok′,Zp(i)), where ok′ is the ring of p-integers of k. We give conditions for the inclusion Шe´t2(k,Zp(i))⊆He´t2(ok′,Zp(i)) to split. We analyze this problem using Iwasawa theory. In particular we relate this splitting problem to the triviality of two invariants, namely the asymptotic kernels of the Galois descent and codescent for class groups along the cyclotomic tower of k. We illustrate our results in both split and non-split cases for quadratic number fields.

Reflection Theorems and the Tame Kernel of a Number Field

Let L be a number field containing the pth primitive root of unity ζ p . We investigate the p-rank of the ideal class groups of some subfields of L by using reflection theorems and establish relations between the p-rank of the ideal class groups and that of groups of units of some subfields of L. Let F be a number field and 𝒪 F the ring of integers in F. We also study the p-rank of tame kernels of F and establish relations between the p-rank of K 2𝒪 F and that of some direct summands of the ideal class group of F(ζ p ).

Tame Kernels of Quaternion Number Fields

Let E/F be a Galois extension of number fields with the quaternion Galois group Q 8. In this paper, we prove some relations connecting orders of the odd part of the kernel of the transfer map of the tame kernel of E with the same orders of some of its subfields. Let E/ℚ be a Galois extension of number fields with the Galois group Q 8 and p an odd prime such that p ≡ 3 (mod 4). We prove that if there is at most one quadratic subfield such that the p-Sylow subgroup of the tame kernel is nontrivial, then p r -rank(K 2(E/K)) is even, i.e., 2|p r -rank(K 2(𝒪 E )) − p r -rank(K 2(𝒪 K )), where K is the quartic subfield of E.

Tame kernels and second regulators of number fields and their subfields

AbstractAssuming a version of the Lichtenbaum conjecture, we apply Brauer-Kuroda relations between the Dedekind zeta function of a number field and the zeta function of some of its subfields to prove formulas relating the order of the tame kernel of a number fieldFwith the orders of the tame kernels of some of its subfields. The details are given for fieldsFwhich are Galois over ℚ with Galois group the group ℤ/2 × ℤ/2, the dihedral groupD2p;pan odd prime, or the alternating groupA4. We include numerical results illustrating these formulas.

数域中tame 核的<italic>p</italic>-秩

Let <italic>E/F</italic> be a Galois extension of number fields with Galois group <italic>G</italic>=Gal(<italic>E/F</italic>), and let <italic>p</italic> be a prime not dividing #<italic>G</italic>. In this paper, using character theory of finite groups, we obtain the upper bound of #<italic>K</italic>₂<italic>O_E</italic> if the group <italic>K</italic>₂<italic>O_E</italic> is cyclic, and prove some results on the divisibility of the <italic>p</italic>-rank of the tame kernel <italic>K</italic>₂<italic>O_E</italic>, where <italic>E/F</italic> is not necessarily abelian. In particular, in the case of <italic>G</italic>=<italic>C_n, D_n, A</italic>₄, we easily get some results on the divisibility of the <italic>p</italic>-rank of the tame kernel <italic>K</italic>₂<italic>O_E</italic> by the character table. Let <italic>E</italic>/Q be a normal extension with Galois group <italic>D_l</italic>, where <italic>l</italic> is an odd prime, and <italic>F</italic>/Q a non-normal subextension with degree <italic>l</italic>. As an application, we show that <italic>f</italic>|<italic>p</italic>-rank <italic>K</italic>₂<italic>O_F</italic>, where <italic>f</italic> is the smallest positive integer such that <italic>p^f</italic>≡±1(mod <italic>l</italic>).

The densities for 3-ranks of tame kernels of cyclic cubic number fields

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(K2OF) ⩽ 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(K2OF) = 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 $$d_{\infty ,r} = 3^{ - r^2 } {{\prod\limits_{k = 1}^\infty {\left( {1 - 3^{ - k} } \right)} } \mathord{\left/ {\vphantom {{\prod\limits_{k = 1}^\infty {\left( {1 - 3^{ - k} } \right)} } {\prod\limits_{k = 1}^r {\left( {1 - 3^{ - k} } \right)} }}} \right. \kern-\nulldelimiterspace} {\prod\limits_{k = 1}^r {\left( {1 - 3^{ - k} } \right)} }}^2 .$$ This suggests that the Cohen-Lenstra conjecture for ideal class groups can be extended to the tame kernels of cyclic cubic number fields.

The 8-rank of tame kernels of quadratic number fields

The 8-rank of tame kernels of quadratic number fields

Odd parts of tame kernels of dihedral extensions

Odd parts of tame kernels of dihedral extensions

Tame kernels of pure cubic fields

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.

On the 3-rank of tame kernels of certain pure cubic number fields

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.

On the odd part of tame kernels of biquadratic number fields

On the odd part of tame kernels of biquadratic number fields

The Tame Kernel of Multiquadratic Number Fields

Let F/K be a Galois extension of a number field of degree n, 𝒪 F the ring of integers in F, and p a prime number which does not divide n. Let K 2 denote the Milnor K-functor. In this article, we shall study the structure of the odd part of the tame kernel K 2𝒪 F of F by using the intermediate fields of F/K. In particular, for a multiquadratic field F, we shall get the p i -rank, (i > 0) of K 2𝒪 F . Finally, we shall determine the structure of the odd parts of K 2𝒪 F when where − 100 < d < 0, d 1 = 2,3,5,7.

Tame kernels of cyclic extensions of number fields

Tame kernels of cyclic extensions of number fields