Extensions Of Algebraic Number Fields Research Articles

Norm relations and computational problems in number fields

For a finite group G $G$ , we introduce a generalization of norm relations in the group algebra Q [ G ] $\mathbb {Q}[G]$ . We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the arithmetic invariants of the subfields of a normal extension of algebraic number fields with Galois group G $G$ . On the algorithmic side, this leads to subfield based algorithms for computing rings of integers, S $S$ -unit groups and class groups. For the S $S$ -unit group computation this yields a polynomial time reduction to the corresponding problem in subfields. We compute class groups of large number fields under GRH, and new unconditional values of class numbers of cyclotomic fields.

The discriminant of compositum of algebraic number fields

For an algebraic number field [Formula: see text], let [Formula: see text] denote the discriminant of an algebraic number field [Formula: see text]. It is well known that if [Formula: see text] are algebraic number fields with coprime discriminants, then [Formula: see text] are linearly disjoint over the field [Formula: see text] of rational numbers and [Formula: see text], [Formula: see text] being the degree of [Formula: see text] over [Formula: see text]. In this paper, we prove that the converse of this result holds in relative extensions of algebraic number fields. We also give some more necessary and sufficient conditions for the analogue of the above equality to hold for algebraic number fields [Formula: see text] linearly disjoint over [Formula: see text].

On the distribution of norm groups in the intervals corresponding to odd degree extensions of algebraic number fields

Let X be a subgroup of a group Y. The interval (X, Y) is the set of subgroups of Y that contain X including X and Y. Let K/k be a finite extension of a -adic number field k. One of the fundamental theorems local class field theory establishes a correspondence between the finite number of norm groups contained in the interval and finite extensions of k. In our earlier work, we proved that iff for any finite Galois extensions of an algebraic number field k. It is natural to determine the norm groups contained in the interval for a given finite extension K/k of algebraic number fields. In our earlier work, we showed that there are extensions K/k such that the corresponding interval contains only a finite number of norm groups, and there are extensions with the corresponding interval containing infinitely many norm groups. The extensions that we considered in our earlier work were primarily of even degrees. In the present work, we investigate the distribution of norm groups in the intervals corresponding to extensions of algebraic number fields of primarily odd degrees divisible by two primes.

On the ideal class groups of the maximal cyclotomic extensions of algebraic number fields

We shall consider the maximal cyclotomic extension of a totally real finite algebraic number field and its ideal class group. We shall investigate the structure of the ideal class group with the action of the cyclotomic Galois group.

Differents, discriminants and Steinitz classes

Let L|K be an extension of algebraic number fields. By a theorem of Hecke, the ideal class of the different 픇L|K is a square in ClL. Whereas the class of the discriminant 픡L|K=NL|K(픇L|K) is the square of the Steinitz class sL|K in ClK, which vanishes if and only if the ring RL of integers in L is a free module over RK, there is no distinguished square root for the different in general. In an attempt to give some interpretation for Hecke's theorem, we examine when there is r∈ClL such that r2=[픇L|K] and NL|K(r)=sL|K (‘Steinitz root’). In this case, every ideal of L in the class r−1 is a free RK-module.

Steinitz classes of tamely ramified Galois extensions of algebraic number fields

The Steinitz class of a number field extension K / k is an ideal class in the ring of integers O k of k, which, together with the degree [ K : k ] of the extension determines the O k -module structure of O K . We call R t ( k , G ) the set of classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. We define A ′ -groups inductively, starting with abelian groups and then considering semidirect products of A ′ -groups with abelian groups of relatively prime order and direct products of two A ′ -groups. Our main result is that the conjecture about realizable Steinitz classes for tame extensions is true for A ′ -groups of odd order; this covers many cases not previously known. Further we use the same techniques to determine R t ( k , D n ) for any odd integer n. In contrast with many other papers on the subject, we systematically use class field theory (instead of Kummer theory and cyclotomic descent).

A formula for Tignol's constant

Let ( K , v ) be a Henselian valued field and ( L , w ) be a finite separable extension of ( K , v ) . In 2004, it was proved that the set A L / K defined by A L / K = { v ( Tr L / K ( α ) ) − w ( α ) | α ∈ L , α ≠ 0 } has a minimum element if and only if [ L : K ] = e f where e , f are the ramification index and the residual degree of w / v , i.e., ( L , w ) / ( K , v ) is defectless. The constant min A L / K was first introduced by Tignol and is referred to as Tignol's constant. In 2005, K. Ota and Khanduja gave a formula for min A L / K when ( L , w ) / ( K , v ) is an extension of local fields. In this paper, we give this formula when ( L , w ) is any finite separable defectless extension of a Henselian valued field of arbitrary rank and thereby generalize some well-known results of Dedekind regarding “different” of extensions of algebraic number fields and ramification of prime ideals.

GEOMETRIC GALOIS THEORY, NONLINEAR NUMBER FIELDS AND A GALOIS GROUP INTERPRETATION OF THE IDELE CLASS GROUP

This paper concerns the description of holomorphic extensions of algebraic number fields. After expanding the notion of adele class group to number fields of infinite degree over ℚ, a hyperbolized adele class group [Formula: see text] is assigned to every number field K/ℚ. The projectivization of the Hardy space ℙ𝖧•[K] of graded-holomorphic functions on [Formula: see text] possesses two operations ⊕ and ⊗ giving it the structure of a nonlinear field extension of K. We show that the Galois theory of these nonlinear number fields coincides with their discrete counterparts in that 𝖦𝖺𝗅(ℙ𝖧•[K]/K) = 1 and 𝖦𝖺𝗅(ℙ𝖧•[L]/ℙ𝖧•[K]) ≅ 𝖦𝖺𝗅(L/K) if L/K is Galois. If K ab denotes the maximal abelian extension of K and 𝖢K is the idele class group, it is shown that there are embeddings of 𝖢K into 𝖦𝖺𝗅⊕(ℙ𝖧•[K ab ]/K) and 𝖦𝖺𝗅⊗(ℙ𝖧•[K ab ]/K), the "Galois groups" of automorphisms preserving ⊕ (respectively, ⊗) only.

Equality of norm groups of subextensions of S n( n⩽5) extensions of algebraic number fields

Let E/ k be a Galois extension of algebraic number fields with the Galois group isomorphic to the symmetric group S n on n⩽5 letters. For any field extensions k⊆ K, L⊆ E a necessary and a sufficient condition is given for the equality N K/kK ∗=N L/kL ∗ to hold, where N K/kK ∗ is the group of norms from K to k of the elements of the multiplicative group K ∗ of K.

On an Obstruction to the Hasse Norm Principle and the Equality of Norm Groups of Algebraic Number Fields

LetL/kandT/kbe finite extensions of algebraic number fields. In the present work we introduce the factor group ofk*∩NL/kJLNT/kJTby (k*∩NT/kJT)NL/kL*, whereJLandJTare the idele groups ofLandT, respectively. The main theorem shows that the computation of this factor group can be reduced to the computation in finite group theory, and the computation with Galois groups of local extensions at a finite number of primes of the base fieldk. We then apply the main theorem to establish a number of interesting results on the equality of norm groups as subgroups of the multiplicative group ofk. In particular, we obtain new results on solitary non-Galois extensions of algebraic number fields.

Galois Module Structure of Ambiguous Ideals in Biquadratic Extensions

AbstractLet N/K be a biquadratic extension of algebraic number fields, and G = Gal(N/K). Under a weak restriction on the ramification filtration associated with each prime of K above 2, we explicitly describe the ℤ[G]-module structure of each ambiguous ideal of N. We find under this restriction that in the representation of each ambiguous ideal as a ℤ[G]-module, the exponent (or multiplicity) of each indecomposable module is determined by the invariants of ramification, alone.For a given group, G, define SG to be the set of indecomposable ℤ[G]-modules, M, such that there is an extension, N/K, for which G ≅ Gal(N/K), and M is a ℤ[G]-module summand of an ambiguous ideal of N. Can SG ever be infinite? In this paper we answer this question of Chinburg in the affirmative.

On a maximal torus in subgroups of a general linear group

Let k be a field, K/k a finite extension of it of degree n. We denote G=Aut(kK), Go=Aut(k K) and fix in K a basis ω1,...,ωn over k. In this basis, to any automorphism group of kK there corresponds a matrix group, which is denoted by the same symbol. Let G′≤G., In this paper, the conditions under which G′⊎Go is a maximal torus in G′ are studied. The calculation of NG′(G′⊎Go) is carried out, provided that thee conditions are fulfilled. The case G′=SL (kK) is of particular interset. It is known that for Galois extensions and for extensions of algebraic number fields, G′⊎Go is a maximal torus in G′. Bibligraphy: 2 titles.

An arithmetic site for the rings of integers of algebraic number fields

The study of extensions with prescribed ramification plays a central role in algebraic number theory. It is a general hope to benefit from the analogy between number fields and function fields of trancendency degree 1 over finite fields, because for the latter the situation is much better understood. In this paper we introduce a new kind of restriction for the ramification of primes in extensions of algebraic number fields and we study the corresponding covering type over the ring of integers. We will prove that the associated fundamental group has a description similar to that of the 6tale fundamental group of a smooth, projective curve over a finite field. Let K be a number field, p be a prime number and let S be a finite set of places of K containing all archimedian places and all places dividing p. Extensions of K which are unramified outside S correspond to &ale covers of the open subscheme Spec(Ox, s) of Spec(Cx). We denote the maximal pro-p Galois extension of K which is unramified outside S by Ks(p) and we write

Relative Galois module structure of integers of abelian fields

Let L/K be an extension of algebraic number fields, where L is abelian over ℚ. In this paper we give an explicit description of the associated order 𝒜 L/K of this extension when K is a cyclotomic field, and prove that o L , the ring of integers of L, is then isomorphic to 𝒜 L/K . This generalizes previous results of Leopoldt, Chan & Lim and Bley. Furthermore we show that 𝒜 L/K is the maximal order if L/K is a cyclic and totally wildly ramified extension which is linearly disjoint to ℚ (m ' ) /K, where m ' is the conductor of K.

Solitary Galois Extensions of Algebraic Number Fields

A finite extension K of an algebraic number field k is called k-solitary if for any finite extension L of k the equality of norm groups N K/k K * = N L/k L * implies that K and L are conjugate over k (i.e., K and L are k-isomorphic). In the present work we investigate finite Galois extensions of k which are k-solitary. We prove that if a finite Galois extension K of k is k-solitary, then K/k is a 2-extension. k and every quadratic extension of k is k-solitary. We establish a necessary condition for a finite Galois 2-extension K/k with ( K : k) ≥ 4 to be k-solitary. We then investigate covering subgroups, and show that the necessary condition is also a sufficient condition for extensions of degree 4 or 8. We conclude our discussion by constructing a Galois extension E of the field of rational numbers Q with the Galois group isomorphic to the dihedral group of order 8 so that E is Q-solitary.

Positively ramified extensions of algebraic number fields.

Positively ramified extensions of algebraic number fields.

On Brauer's induction formula of characters of groups

and with 2~ linear characters of suitable subgroups of G. Here we will show that for any non-principal irreducible character Z a summation 1,) can be found in which none of the 2] contains the principal character 1 a as an irreducible constituent. This answers in the affirmative a like question of G. O. Michler (Essen, Germany), as put to the author in June 1993. As Michler told him, he could not gather an immediate and positive answer to his query from the existing literature, an observation that the author is willing to share with him after loose inspection of known sources. Note, that a positive answer to Michler's question has consequences for positions of occurrency of zeroes of Dedekind ~-functions and of Artin L-functions of algebraic number fields. To be more specific, let E/K be a finite galois extension of algebraic number fields with gatois group G. Consider the Artin L-function L~: = L(s, 2~, E/K). Then if n~ is the order of the zero or pole of L~ at the point s 0, Artin's conjecture maintains that for )~ ~e 1~ ~ a~n~ < 0 holds whenever Z = Z al 2~, a t ~ ~g, is a (*)-decomposition, in which none of the 2~ contains 1 a as irreducible constituent; that is, L(s, Z, E/K) should be analytic at so. For more information about these topics, one may consult ([2], Chapter VIII), ([3]~ Chapter I) and more in particular the references drawn up in [7]; for very recent developments in respect to the Aramata-Brauer theorem one is referred to [6].

On the Index of the Group Generated by Relative Units

Let L/ K be a finite abelian extension of algebraic number fields. We consider the group generated by "relative units" of cyclic subextensions of L/ K. We majorize its index in the full group of units of L.

On the center of Galois groups of maximal pro-p extensions of algebraic number fields with restricted ramification.

On the center of Galois groups of maximal pro-p extensions of algebraic number fields with restricted ramification.

A note on free pro-$p$-extensions of algebraic number fields

For an algebraic number field k and a prime p, define the number p to be the maximal number d such that there exists a Galois extension of k whose Galois group is a free pro-p-group of rank d. The Leopoldt conjecture implies 1~ p ~ r2 +1, (r2 denotes the number of complex places of k). Some examples of k and p with 03C1 = r2 + 1 have been known so far. In this note, the invariant 03C1 is studied, and among other things some examples with p r2 + 1 are given.