Question regarding Galois group of a polynomial.

Question

I am a graduate student.We have Galois theory in this semester.We were first taught splitting fields of a polynomial.Then our instructor introduced the Galois group of a polynomial $f\in F[x]$ to be $Aut_F(E)$ where $E$ is the splitting field of $f$ over $F$ and the set is the collection of all automorphisms on $E$ fixing $F$.Then he gave us some polynomials like $(x^4-2)$ and asked us to find the Galois group upto isomorphism.Now,in no book I find the same definition,some books define Galois group for a field extension.I am confused.So can someone tell me a book that follows this definition?Also can someone tell me how to determine the Galois group,I mean how to find a known group to which the Galois group is isomorphic?Is there any specific method or it is not always so easy to determine the Galois group of a polynomial.

Answer 1

Definition of Galois group is much more general than your instructor's one. The Galois Group is defined for field extensions $L|K$, so the Galois group of the extension is the collection of all automorphisms of L that fix every element in K. For the specific case of an field extension that is a splitting field of some polynomial you have the definition of your instructor. Usually you will get required to find the Galois group of an Galois extension, so a good beginning in this case will be looking for the groups of order the degree of the extension. You will prove that every Galois group of order n is a subgroup of the symmetric group of order n!