Number Theory/Quadratic Fields

Introduction

A quadratic field F is an extension of degree 2 of the rational number field Q. The solution of the quadratic equation shows that it is generated over Q by the square root of a rational number. Multiplying (or dividing) this rational number by the square of a suitable integer, we may assume that it is a “square free” integer m, that is, an integer m without square factors. Let F = Q(s) where s.s = m. Any element x of F is of the form x = a + bs with a and b in Q. The mapping a + bs --> a – bs is an automorphism of F over Q. Thus F is a normal extension of Q (with a cyclic group of order 2 as Galois group).