Developments in the theory of algebras over number fields: A new foundation for the Hasse norm residue symbol and new approaches to both the Artin reciprocity law and class field theory

2011 ◽  
pp. 117-151
Keyword(s):  
Field Theory ◽  
Number Fields ◽  
Class Field Theory ◽  
Class Field ◽  
Residue Symbol ◽  
New Approaches ◽  
Reciprocity Law ◽  
Norm Residue

Using decomposition groups to prove theorems about quadratic residues

2020 ◽  
Vol 1 (1) ◽  
pp. 12-20
Author(s):  
Tomas Perutka
Keyword(s):  
Field Theory ◽  
Number Theory ◽  
Class Field Theory ◽  
Legendre Symbol ◽  
Quadratic Reciprocity ◽  
Class Field ◽  
Reciprocity Laws ◽  
Reciprocity Law ◽  
Quadratic Residues ◽  
Cyclotomic Extensions

In this text we elaborate on the modern viewpoint of the quadratic reciprocity law via methods of alge- braic number theory and class field theory. We present original short and simple proofs of so called addi- tional quadratic reciprocity laws and of the multiplicativity of the Legendre symbol using decompositon groups of primes in quadratic and cyclotomic extensions of Q.

scholarly journals On 2-class field towers for quadratic number fields with 2-class group of type (2,2)

1998 ◽  
Vol 40 (1) ◽  
pp. 63-69 ◽  
Author(s):  
Frank Gerth
Keyword(s):  
Field Theory ◽  
Class Number ◽  
Class Group ◽  
Number Fields ◽  
Class Field Theory ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Class Field ◽  
Class Field Tower

Let K be a quadratic number field with 2-class group of type (2,2). Thus if Sk is the Sylow 2-subgroup of the ideal class group of K, then Sk = ℤ/2ℤ × ℤ/2ℤ LetK ⊂ K1 ⊂ K2 ⊂ K3 ⊂…the 2-class field tower of K. Thus K1 is the maximal abelian unramified extension of K of degree a power of 2; K2 is the maximal abelian unramified extension of K of degree a power of 2; etc. By class field theory the Galois group Ga1 (K1/K) ≅ Sk ≅ ℤ/2ℤ × ℤ/2ℤ, and in this case it is known that Ga(K2/Kl) is a cyclic group (cf. [3] and [10]). Then by class field theory the class number of K2 is odd, and hence K2 = K3 = K4 = …. We say that the 2-class field tower of K terminates at K1 if the class number of K1 is odd (and hence K1 = K2 = K3 = … ); otherwise we say that the 2-class field tower of K terminates at K2. Our goal in this paper is to determine how likely it is for the 2-class field tower of K to terminate at K1 and how likely it is for the 2-class field tower of K to terminate at K2. We shall consider separately the imaginary quadratic fields and the real quadratic fields.

A Generalization of Global Class Field Theory

1969 ◽  
Vol 21 ◽  
pp. 609-614
Author(s):  
Tae Kun Seo ◽  
G. Whaples
Keyword(s):  
Field Theory ◽  
Finite Field ◽  
Class Group ◽  
Rational Functions ◽  
Class Field Theory ◽  
Usual Sense ◽  
Finite Degree ◽  
Class Field ◽  
Reciprocity Law

Let R be a field of rational functions of one variable over a field of constants R0. Dock Sang Rim (6) has proved that the global reciprocity law in exactly the usual sense holds whenever R0 is an absolutely algebraic quasi-fini te field of characteristic not equal to 0: this was known before only when R0 was a finite field. We shall give another proof of Rim's result by means of a noteworthy generalization of the usual global reciprocity law. Namely, let R0 be a finite field and let F be the set of all fields k contained in some fixed Ralg.clos. and of finite degree over R. The reciprocity law states that there exists a family {fk}, k ∈ F, of functions fk: Ck → G(kabel.clos./k) (where Ck is the idèle class group of k) enjoying certain properties such as the norm transfer law.

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