scholarly journals How Hasse was led to the Theory of Quadratic Forms, the Local-Global Principle, the Theory of the Norm Residue Symbol, the Reciprocity Laws, and to Class Field Theory

2018 ◽  
Author(s):  
Günther Frei
Keyword(s):  
Field Theory ◽  
Quadratic Forms ◽  
Class Field Theory ◽  
Class Field ◽  
Residue Symbol ◽  
Reciprocity Laws ◽  
Norm Residue ◽  
Global Principle

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.

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