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/? 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.     

The Structure of the Tame Kernels of Quadratic Number Fields (III)

Let F be an imaginary quadratic number field and K ₂ O _F the tame kernel of F. In this article, we determine all possible values of r ₄(K ₂ O _F ) for each type of imaginary quadratic number field F. In particular, for each type of imaginary quadratic number field we give the maximum possible value of r ₄(K ₂ O _F ) and show that each integer between the lower and upper bounds occurs as a value of the 4-rank of K ₂ O _F for infinitely many imaginary quadratic number fields F.     

On the p 2-Rank of Tame Kernels of Quadratic Fields

For any odd prime p, we prove some results connecting the p²-rank of the tame kernel of a quadratic field F with the p²-rank Cl(𝒪_E1 ), where E₁ is the maximal real subfield of F(ζ_p² ).     

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.     

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 ).     

关于虚二次域Tame核计算的一个注记

本文给出了一种计算数域Tame核的方法.应用到虚二次域上,证明了当 Nv>8δD6时,(？)t／:K2S'F／K2SF→k*是双射。     

A Remark on Computing the Tame Kernel of Quadratic Imaginary Fields

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 is bijective when . Received October 25, 1999, Accepted February 5, 2001     

Bi-Quadratic Number Fields with Trivial 2-Primary Hilbert Kernels

Using results of Browkin and Schinzel one can easily determinequadratic number fields with trivial 2-primary Hilbert kernels.In the present paper we completely determine all bi-quadraticnumber fields which have trivial 2-primary Hilbert kernels.To obtain our results, we use several different tools, amongstwhich is the genus formula for the Hilbert kernel of an arbitraryrelative quadratic extension, which is of independent interest.For some cases of real bi-quadratic fields there is an ambiguityin the genus formula, so in this situation we use instead Brauerrelations between the Dedekind zeta-funtions and the Birch–Tateconjecture. 2000 Mathematics Subject Classification 11R70, 19F15.     

Tame and wild kernels of quadratic imaginary number fields

For all quadratic imaginary number fields of discriminant
we give the conjectural value of the order of Milnor's group (the tame kernel) where is the ring of integers of Assuming that the order is correct, we determine the structure of the group and of its subgroup (the wild kernel). It turns out that the odd part of the tame kernel is cyclic (with one exception, ).

     

Roots of Irreducible Polynomials in Tame Henselian Extension Fields

A class of irreducible polynomials 𝒫 over a valued field (F, v) is introduced, which is the set of all monic irreducible polynomials over F when (F, v) is maximally complete. A “best-possible” criterion is given for when the existence of an approximate root in a tamely ramified Henselian extension K of F of a polynomial f in 𝒫 guarantees the existence of an exact root of f in K.     

Tame Automorphisms of Elementary Free Groups

We prove that the automorphism group of a finitely generated fully residually free group is tame.     

On Tame Kernels and Ideal Class Groups

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）.     

Imaginary Quadratic Fields with Ono Number 3

In this article, we prove that an imaginary quadratic field F has the ideal class group isomorphic to ?/2? ⊕ ?/2? if and only if the Ono number of F is 3 and F has exactly 3 ramified primes under the Extended Riemann Hypothesis (ERH). In addition, we give the list of all imaginary quadratic fields with Ono number 3.     

关于一些二次数域的Tame核的2-Sylow子群

本文主要研究二次数域F=φ()的2-sylow子群,其中d只有两个不同奇素数和2∈NF/φ(F*).建立了E=φ()的类群和具有平凡的4-秩或8-秩的K2OF两者之间等价关系.     

On Tame Kernel and Class Group in Terms of Quadratic Forms

The paper is to investigate the structure of the tame kernel K₂O_F for certain quadratic number fields F, which extends the scope of Conner and Hurrelbrink (J. Number Theory88 (2001), 263-282). We determine the 4-rank and the 8-rank of the tame kernel, the Tate kernel, and the 2-part of the class group. Our characterizations are in terms of binary quadratic forms X²+32Y,X²+64Y²,X²+2Py²,2X²+Py²,X²−2Py²,2X²−Py². The results are very useful for numerical computations.     

Tame group actions on central simple algebras

We study actions of linear algebraic groups on finite-dimensional central simple algebras. We describe the fixed algebra for a broad class of such actions.     

Tame roots of wild quivers

We study the tame behaviour of the representations of wild quivers Q via tame roots. A positive root d of Q is called a tame root if d is sincere and for any positive sub-root d^′ of d we have q(d^′)0, where q(d^′) is the Tits form of Q. We prove that a sincere root is a tame root if and only if for any decomposition of d into a sum of positive sub-roots d=d¹++d^s, there is at most one dⁱ with q(dⁱ)=0 and q(d^j)=1. This is the essential property of a tame root and it is an alternative way to define tame roots. Then we give the canonical decomposition of a tame root. At the end we prove our main result that for any wild graph, there are only finitely many tame roots.