ON THE RELATION BETWEEN THE GALOIS GROUP OF AN EQUATION AND ITS DISCRIMINANT

Abstract

In the algebra we often from an extension field (or-ring, integral domain) out of a given subfield (or-ring, integral domain), and its algebraic property is not changed from that of the subfield (or-ring, -integral domain). Forming a new field (or-ring, integral domain) out of the previously given subfield (or-ring, -integral domain) is often used in the modern algebra and is called the principle of extension. An extension field ∑ of K is called an algebraic extension over K if every element of ∑ is algobraic over K. If ∑ is an algesraic extension field of a commutative field K and the aplitting field over K of an arditrary irreduceible polynomial f(x) in K, is always included in ∑, then ∑ will be said the Galois extension over K. If ∑ is a separable Galois extension of degree n over K, then there exist n automorphisms oh ∑/K. These form a group with respect to the multiplication of the automorphisms. This group made up of the all autonorphisms of ∑/K is the Galois group of ∑/K. In this thesis, I am going to comsider a relation between the Galois group of an equation and it's diacriminant, which is wellknown in the Theory of Equations and states as follows; Proposition (1) The group of an equation consists of even permutations alone if and only if the aquare root of the discriminant is contained in the base fileld. Otherwide only half of its permutations are even.