| Summary: | 碩士 === 國立中正大學 === 數學研究所 === 102 === Suppose given a finite group G.
Is it always possible to find a polynomial f(t) over Q such that the Galois group of f(t) over Q is isomorphic to G? This is the so-called Inverse Galois Problem.
In Section 1.1, we list some preliminary propositions and results in algebra.
In Section 1.2, we introduce the discriminant of a polynomial f(t).
This is a tool to help us to determine the Galois group of f(t) over a field K.
Note that the Galois group of an irreducible polynomial of degree n is isomorphic to a transitive subgroup in Sn.
In Chapter 2, we will study the generators of Sn and An.
There is a special case for Sp when p is a positive prime integer.
In Chapter 3, we will find all the transitive subgroups in Sn for n=3,4,5.
This is a long procedure in finding these groups.
In Chapter 4, we will introduce a special subgroups GA(p) in Sp where p is a positive prime integer. This group GA(p) is a transitive and solvable group in Sp with order p(p-1).
Moreover, a group G is a transitive and solvable subgroup of Sp if and only if G is conjugate to a subgroup of GA(p).
We let $t$ be an indeterminant over a field K in Chapter 5 and Chapter 6.
In Chapter 5, we will study the cubic polynomials f(t) over K.
We will use this result in Chapter 6.
In Chapter 6, we will find examples of irreducible polynomials whose Galois groups are transitive subgroups of Sn for n=3, 4, 5.
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