Number Field Of Degree Research Articles

On the Absolute Ideal Class Groups of Relatively Meta-Cyclic Number Fields of a Certain Type

Notations. The following notations will be used throughout this paper.˛: the identity of a finite group.Q: the rational number field.P: an algebraic number field of finite degree, fixed as the ground field.l: a prime number.ζl: a primitive l-th root of unity.

On Meta Abelian Fields of a Certain Type

Let k be an algebraic number field of finite degree, and l a rational prime (including 2); k and l being fixed throughout this paper. For any power ln of l, denote by ζn an arbitrarily fixed primitive ln-th root of unity, and put kn = k(ζn). Let r be the maximal rational integer such that ζr∈k i.e. kr = k and kr+1≠k.

An application of the generalized norm residue symbol

Let k be an algebraic number field of finite degree or an algebraic function field of dimension 1 with finite constant field and let K/k be a finite, separably Galois extension field. We call an algebraic extension field L, not necessarily Galois nor necessarily of finite degree, of K an everywhere locally abelian extension, denoted in the following as EL-abelian extension, for brevity, of K over k, if and only if for every finite extension L' of K in L and every valuation v of L it holds that the completion field Lv' of L' with reference to v is a composition field of KV and an abelian extension Mv of kv in Lv', where we denote by kv and KV the closures of k and K in Lv', respectively. Let Q be an algebraic closure of K, BK/k the maximum separably EL-abelian extension of K/k in Q, CK/k the maximum separably central extension of K/k in Q (i.e. the maximum extension CK/k which is separably normal over k, contains K, and has the Galois group of CK/k/K in the center of the Galois group of CK/k/k), Ik and IK the groups of the ideles of k and K, Ak and AK the maximum separably abelian extensions of k and K in Q, respectively. Let DK/k=BK/k nCKIk. Then, the norm residue symbol 0Jk in Ak/k is, if restricted into NK/kIK, extensible into a homomorphism UK/k of NK/kIK into the Galois group G(DK/k/K). We shall study in the present article arithmetical meanings of the extensibility. In ?1, we shall define oUK/k precisely, in ?2, study the kernel and obtain a principal genus theorem (Theorem 2), from which will follow easily a certain generalized formulation of some fundamental theorems in the class field theory.1 Combining it with a known property2 of the quotient group of the connected component of the unit element of the idele class group by the natural image of the maximal compact subgroup of the connected component of the unit element of the idele group, we shall obtain, in ?3, our main result (Theorem 8), which concerns total norm residues in the principal idele group of k for K/k, and which is, the author thinks, an idele-theoretic reconstruction of a research of Scholz.3

Unit Groups of Cyclic Extensions

LetΩbe an algebraic number field of finite degree, which we fix once for all, and letKbe a cyclic extension overΩsuch that the degree ofK/Ωis a powerof a prime numberl. It is obvious that the norm groupNK/ΩeKof the unit group ekofK, being a subgroup of the unit group e of Ω contains the groupconsisting of all-th powersof ε∈e.

Cubic forms over algebraic number fields

It is our purpose here to show that, using results already in the literature, it is easy to prove the following and similar theorems. For every positive integer d, there exists an integer Ψ ( d ) such that if K is an algebraic number field of degree d over the field of rational numbers then every cubic form f(x 1 x 2 , …, x n ) over K, with n ≥ Ψ(d), has a non-trivial zero in K.

On the Ramification Theory of Infinite Algebraic Extensions

The ramification theory of the infinite normal algebraic extensions was first treated by J. Herbrand and C. Chevalley [5] in 1933. The Hilbert's ramification groups Q3, of an infinite normal algebraic extension K/k were defined there as the projective limit groups Q~x = proj. lim 58n) of the corresponding ramification groups Q!n) of intermediate finite normal extensions k/Ik; k C ki C : c kn c , K Uw~lkn . It was proved there that the set A of all the ramification groups IQA} are linearly ordered by inclusion and A is metrically complete by a suitable metric topology. It was also proved that all the ramification groups of a normal subfield K'/k (k c K' C K) and of K/k' (k c k' c K) are induced from that of K/k. Since then no remarkable developments and applications of this theory have been made.' Recently T. Tamagawa [11] constructed the conductor theory for Weil's algebroid character and proved a functional equation for the Weil's L-function. He used there essentially the Hilbert's ramification theory of abelian and other related infinite extensions over an algebraic number field of finite degree. Associated with these investigations of Tamagawa, I. Satake [10] also constructed the Hilbert's ramification theory of some types of infinite algebraic extensions, which he called the (H)-extensions, over a field with a discrete or semi-discrete valuation. The purpose of this paper is also to investigate the ramification theory of general infinite separable normal algebraic extensions over a field k with a discrete valuation whose residue class field is perfect. In ?2 we shall give necessary preparations for finite extensions. In ?3 we shall introduce a suitable parameter it for the ramification fields (Theorem 3. A). For this sake we shall define a real canonical set U(K/k) and derive from it two sets S(K/k) and S*(K k;) such that S*(K/k) _ U(K/k) C S(K/k). We shall then classify the ramification fields into two kinds. For the ramification fields of the first kind bKlk(u) we shall introduce a real parameter u e S(K/k), and for that of the second kind UK/k(u) we shall introduce a real parameter ue S*(K/k). We can determine the inclusion and limit relations between these ramification fields (Proposition 3.3, 3.5). The relation between these ramification fields of K/k and of a normal subfield K'/k (or of K/Tk', k c k' c K) can be explicitly given by our parameter u (Theorem 3.B). In ?4 we shall prove a Theorem of Tamagawa [11] (Proposition 4.3) as an application of our theory, and construct some examples of K/k for which the canonical set U(K/k) has the given properties. In ?5, ?6 we shall define the p-exponents of the different [bo(Klk)] and Artin's

On a 3-Cohomology Class in Class Field Theory and the Relationship of Algebra- and Idele-Classes

Let k be an algebraic number field (of finite degree) and K/k be a (finite) Galois extension with Galois group R. In connection with the class field theory, Weil [101 derived a certain canonical factor set a(o, r) of 9 in the idele-class group (K of K, which is uniquely determined by K/k in the sense of equivalence. For instance, a (mod R') -?f t r(o, A) (Ek k, essentially) (mod NK/k( RK)) is the Artin reciprocity law isomoprhism of 9/9' and Sk/NK/k( SK), where 9' is the commutator subgroup of R. A different approach to this (idele-class) factor set was given in [6]. Let, for each pair o, r, a(o, r) be a representative idele of a(o, r). Then (3a)(p, o, r) = a(o, r)a(po, r)-'a(p, o-r)a(p, oY)is, for each triple p, a, r, a principal idele, i.e. a non-zero element of K. We obtain thus a 3-cochain a(p, a, r) = (6a)(p, a, r) of 9 in the principal idele group PK of K, which is in fact a 3-cocycle. We saw, in [6], that this 3-cycocle a, or more precisely its cohomology class, attached in invariant fashion to the Galois extension K/k, has the order m/m', where m = (K: k) and m' is the least common multiple of the p-degrees of K/k, p running over all primes in k. It was shown further, in a joint paper [4] by Hochschild and the writer, that the cyclic group, of order m/m', generated by this 3-cohomology class is exactly the group of all 3-cohomology classes of 9 in PK with ceiling. This last group is in turn, according to EilenbergMacLane [1], nothing but the group of all Teichmtiller 3-cohomology classes for K/k, belonging to central simple algebras over K to which every element of ft can be extended (as automorphism). Now arises the problem to determine the exact algebra-class (though not unique) which possesses our a, attached to K/k, as its Teichmiiller-class. The present work is devoted to this problem and our solution is given in our Theorem in ?3. If a central simple algebra A over K, to which every automorphism of K/k can be extended, is given in a crossed-product form for instance, we can obtain its Teichmiiller-cocycle in a more or less explicit process involving the HilbertSpeiser theorem. On the other hand, in order to obtain our 3-cocycle a, we repeat the construction of the canonical idele-class factor set a given in [6], which is nothing but the global analogy of the local construction in Hochschild [3]1 and makes use of the idele interpretation of the principal genus theorem of Noether, but with more detailed analysis. Namely, we carry out all the processes in terms of ideles, rather than of idele-classes, preserving thus all principal idele factors which were discarded in [6]. This construction, involving the theorem of everywhere splitting algebras, the Hilbert-Speiser theorem and its idble analogy, leads to a. Though fairly explicit, these constructions of a and the Teichmfiller-

On uniform distribution of algebraic numbers

Let K be an algebraic number field of degree n + 1 over the rational field R. The conjugates of K are denoted by K(?), , I K(n), it being supposed that K M K(cl are real, K(r?l+), * , K(n) are complex, and moreover that for k =r1+ *, ri +r2 the fields K(k) and K(k+r2) are obtainable from one another by interchanging i( = (1)1/2) and -i (here r1+2r2=n). I assume throughout that r1 ?0 and that n ?1. Numbers in K are represented by small Greek letters, the insertion of superscripts denoting the passage to the corresponding conjugates. Let co, * * , wn be a set of numbers in K which are linearly independent over R; let co = 1; and finally let f(x1, * * *, xn) be a (complex-valued) function of n real variables which is Riemann integrable over the unit cube En: O<Xk<l (k=1, * * *, n). Then, according to theorems of Weyl on uniform distribution,