scholarly journals Annihilator ideals of two-generated metabelian p-groups

2018 ◽  
Vol 17 (04) ◽  
pp. 1850076
Author(s):  
Daniel C. Mayer
Keyword(s):  
Additive Group ◽  
Number Fields ◽  
Galois Groups ◽  
Algebraic Number Fields ◽  
Bivariate Polynomials ◽  
Integer Coefficients ◽  
Commutator Formula ◽  
Class Fields ◽  
Residue Class Ring ◽  
Class Ring

For certain metabelian [Formula: see text]-groups [Formula: see text] with two generators [Formula: see text] and [Formula: see text], the annihilator [Formula: see text] of the main commutator [Formula: see text] of [Formula: see text], as an ideal of bivariate polynomials with integer coefficients, is determined by means of a presentation for [Formula: see text]. It is proved that together with Schreier’s polynomials [Formula: see text], the annihilator [Formula: see text] identifies the group [Formula: see text] uniquely, and the Furtwängler isomorphism of the additive group underlying the residue class ring [Formula: see text] to the commutator subgroup [Formula: see text] of [Formula: see text] admits the calculation of the abelian type invariants of [Formula: see text]. The results are underpinned by class field theoretic realizations of the groups [Formula: see text] as Galois groups [Formula: see text] of second Hilbert [Formula: see text]-class fields [Formula: see text] over algebraic number fields [Formula: see text].

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

2005 ◽  
Vol 16 (06) ◽  
pp. 567-593
Author(s):  
T. M. GENDRON ◽  
A. VERJOVSKY
Keyword(s):  
Galois Theory ◽  
Class Group ◽  
Number Fields ◽  
Field Extension ◽  
Abelian Extension ◽  
Galois Groups ◽  
Algebraic Number Fields ◽  
Infinite Degree ◽  
Groups Of Automorphisms ◽  
Holomorphic Extensions

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.

scholarly journals Kronecker classes of fields and covering subgroups of finite groups

1994 ◽  
Vol 57 (1) ◽  
pp. 17-34 ◽  
Author(s):  
Cheryl E. Praeger
Keyword(s):  
Group Theory ◽  
Finite Groups ◽  
Algebraic Number ◽  
Number Fields ◽  
Galois Groups ◽  
Algebraic Number Fields ◽  
Covering Property ◽  
Bounded Degree ◽  
Field Extensions ◽  
Equivalent Field

AbstractKronecker classes of algebraci number fields were introduced by W. Jehne in an attempt to understand the extent to which the structure of an extension K: k of algebraic number fields was influenced by the decomposition of primes of k over K. He found an important link between Kronecker equivalent field extensions and a certain covering property of their Galois groups. This surveys recent contributions of Group Theory to the understanding of Kronecker equivalence of algebraic number fields. In particular some group theoretic conjectures related to the Kronecker class of an extension of bounded degree are explored.

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