Extensions Of Quadratic Fields Research Articles

Maximal unramified extensions of imaginary quadratic number fields of small conductors

Dans l'article [15], nous donnions dans une table la structure des groupes de Galois Gal(Kur/K) des extensions maximales non ramifiees Kur des corps de nombres quadratiques imaginaires K de conducteur ≤ 1000 sous l'Hypothese de Riemann Generalisee, sauf pour 23 d'entre eux (tous de conducteur ≥723). Ici nous mettons a jour cette table, en precisant, pour 19 de ces <corps exceptionnels, la structure de Gal(K ur /K). En particulier pour K = Q(√-856), nous obtenons Gal(K ur /K) S4 x C 5 et K ur = K 4, le quatrieme corps de classes de Hilbert de K. C'est le premier exemple d'un corps de nombres dont la tour de corps de classes est de longueur 4.

Chinburg's Third Invariant for Abelian Extensions of Imaginary Quadratic Fields

In this paper the authors provide strong evidence for a conjecture of Chinburg in the context of abelian extensions N/K of quadratic imaginary fields K with group G. For abelian groups G, the conjecture specialises to the statement that Chinburg's third invariant Ω(N/K, 3)=0 ∈ Cl(Z[G]). We prove (Theorem 1) that if K has class number 1 (and some technical conditions on the conductor of N/K and its roots of unity hold) then Ω(N/K, 3) lies in the kernel group D(Z[G]). For a special class of genus extensions of prime power degree we prove (Theorem 2) the full conjecture, that Ω(N/K, 3)=0. The proof technique for Theorems 1 and 2 uses Kolyvagin's Euler systems of units applied to certain explicit groups of elliptic units, which essentially determines the difference in Galois structure between units and ideal class groups over the maximal order MG ∈ Q[G]. This is made precise by using Hecke factorisation and working in the factorisability defect class group of the category of finitely generated Z[G]-modules, which is isomorphic to Cl(MG). For Theorem 2, the fact that the class number of N is coprime to [N:K] in such genus extensions N/K allows one to lift these results to Cl(Z[G]), after using cohomological calculations to check that (essentially) an explicit representative for a Tate sequence defining Ω (N/K, 3) has the correct extension class (which is related to the fundamental class of N/K). 1991 Mathematics Subject Classification: 11R33.

The Maximal Unramified Extensions of the Imaginary Quadratic Number Fields with Class Number Two

The maximal unramified extensions of the imaginary quadratic number fields with class number two are determined explicitly under the Generalized Riemann Hypothesis.

Densities for 4-Class Ranks of Totally Complex Quadratic Extensions of Real Quadratic Fields

In this paper we specify density results for the 4-class ranks of totally complex quadratic extensions K of certain real quadratic fields F. Whereas previous density results have been confirmed to cases where the base field F has odd class number, this paper considers various F with nontrivial 2-class groups and even with arbitrarily large 2-class ranks.

Spheres of quadratic field extensions

Spheres of quadratic field extensions

The Exponent 2-Class-Group Problem for Non-Galois Over Q Quartic Fields That Are Quadratic Extensions of Imaginary Quadratic Fields

Focusing on a particular case, we will show that one can explicitly determine the quartic fields K that have ideal class groups of exponent ≤ 2, provided that K/ Q is not normal, provided that K is a quadratic extension of a fixed imaginary quadratic number field, and provided that the regulator of K is not too large compared with the discriminant of K.

Capitulation in unramified quadratic extensions of real quadratic number fields

Letkbe an algebraic number field andCkits ideal class group (in the wider sense). SupposeKis a finite extension ofk. Then we say that an ideal class ofk capitulatesinKif this class is in the kernel of the homomorphisminduced by extension of ideals fromktoK(See Section 2 below). In [4], Iwasawa gives examples of real quadratic number fields,with distinct primesPi≡ 1 (mod 4), for which all the ideal classes of the 2-class group,Ck,2(the 2-Sylow subgroup ofCk), capitulate in an unramified quadratic extension ofk. In these examples,Ck,2is abelian of type (2,2), i.e. isomorphic to ℤ/2ℤ×ℤ/2ℤ and so all four ideal classes capitulate.

Chinburg's Third Invariant in the Factorisability Defect Class Group

AbstractChinburg's third invariant Ω(N/K, 3) ∊ C1(Z[Γ]) of a Galois extension N/K of number fields with group Γ is closely related to the Galois structure of unit groups and ideal class groups, and deep unsolved problems such as Stark's conjecture.We give a formula for Ω(N/K, 3) modulo D(ZΓ) in the factorisability defect class group, reminiscent of analytic class number formulas. Specialising to the case of an absolutely abelian, real field N, we give a natural conjecture in terms of Hecke factorisations which implies the vanishing of the invariant in the defect class group.We prove this conjecture when N has prime-power conductor using Euler systems of cyclotomic units, Ramachandra units and Hecke factorisation. This supports a general conjecture of Chinburg, which in our situation specialises to the statement that Ω(N/K, 3) = 0 for such extensions.We also develop a slightly extended version of Euler systems of units for general abelian extensions, which will be applied to abelian extensions of imaginary quadratic fields elsewhere

Spreads of right quadratic skew field extensions

LetL/K be a right quadratic (skew) field extension and let be a 3-dimensional projective space overK which is embedded in a 3-dimensional projective space overL. Moreover, let ℐ be a line of which carries no point of. The main result is that — even whenL orK is a skew field — the following holds true: A Desarguesian spread of is given by the set of all lines of which are indicated by the points of ℐ. A spread of arises in this way if, and only if, there exists an isomorphism ofL onto the kernel of the spread such thatK is elementwise invariant. Furthermore, a geometric characterization of right quadratic extensions with a left degree other than 2 and of quadratic Galois extensions is given.

Quadratic field extensions and residue homomorphisms of Witt rings

Quadratic field extensions and residue homomorphisms of Witt rings

Elliptic units of cyclic unramified extensions of complex quadratic fields

Elliptic units of cyclic unramified extensions of complex quadratic fields

Witt equivalence classes of quartic number fields

It has recently been established that there are exactly seven Witt equivalence classes of quadratic number fields, and then all quadratic and cubic number fields have been classified with respect to Witt equivalence. In this paper we have classified number fields of degree four. Using this classification, we have proved the Conjecture of Szymiczek about the representability of Witt equivalence classes by quadratic extensions of quadratic fields.

KT-nearfields of rank 2

Nearfields of rank 2 over their kernel which are not Dickson nearfields have recently been constructed by H. Zassenhaus. Here it is shown that KT-nearfields of rank 2 over their kernels and of characteristic ≠ 2 are necessarily Dickson nearfields which are coupled to quadratic field extensions and their Dickson group is isomorphic to ℤ2. Using results of Groger one obtains all KT-nearfields of rank 2 over the field of rational numbers.

The 4-class ranks of quadratic extensions of certain real quadratic fields

Let F be a real quadratic extension of Q in which exactly one prime ramifies. Let K be a quadratic extension of F, and let R K denote the 4-class rank of K. In this paper we specify how likely it is that R K = 0, 1, 2, …. The formulas we obtain are analogous to formulas conjectured by Cohen and Martinet in the prime-to-2-part of the ideal class group of K.

The $4$-class ranks of quadratic extensions of certain imaginary quadratic fields

The $4$-class ranks of quadratic extensions of certain imaginary quadratic fields

Local Orders, Root Numbers, and Modular Curves

Let F be a locally compact field, and let wr be an irreducible, admissible representation of GL2(F) on an infinite dimensional complex vector space. Let c be the central quasicharacter of wx. Let K be a separable quadratic field extension of F, and let X be a quasicharacter of K* such that xw = 1 on F*. The character X gives a 1-dimensional representation of the Weil group WK of K, via the isomorphism Wab K* of local classfield theory. Let -xw be the admissible irreducible representation of GL2(F) which corresponds (in the sense of Langlands) to the 2-dimensional induced representation IndWFX of the Weil group of F. The representation

Class number restrictions for certain $l$-extensions of imaginary quadratic fields

Class number restrictions for certain $l$-extensions of imaginary quadratic fields

The Computational Complexity of Division in Quadratic Extension Fields

We show that, for any quadratic field extension $k \subset K$, the division in K requires exactly six essential multiplications and divisions over k. In particular, complex division in real arithmetic requires six multiplications and divisions. If additions and subtractions are counted as well, exactly nine operations are required if $k \subset K$ is a radical extension; otherwise ten operations are required.

Elliptic units and a Kummer's criterion for imaginary quadratic fields

The article discusses a criterion for the existence of certain cyclic extensions of prime degree p of abelian extensions of imaginary quadratic fields. The condition in the criterion is given in terms of special values of L-functions with Grössencharacters. The method is based on the theory of elliptic units by Siegel, Ramachandra and Robert.

Ray class field extensions of real quadratic fields and solvability of congruences

An explicit description is given of a process using classical class field theory to generate solvability criteria for a class of fourth degree congruences. The method involves finding generators and determining conductors for relatively quadratic extensions of a real quadratic base field. Several examples are given.