Thread: Confusion about splitting fields

I'm studying for an exam and solving some problems from the book on my own. I came across a question about splitting fields which confused me.

Let K be a finite extension of F. Prove that K is a splitting field over F if and only if every irreducible polynomial in F[x] that has a root in K splits completely in K[x].

My question is how they have defined splitting field here. The book up to now always defined it as a field that is associated with a specific polynomial , but this problem doesn't seem to. How can I just prove that K is a splitting field? Over which polynomial should it be a splitting field?

Originally Posted by eeyore

I'm studying for an exam and solving some problems from the book on my own. I came across a question about splitting fields which confused me.

Let K be a finite extension of F. Prove that K is a splitting field over F if and only if every irreducible polynomial in F[x] that has a root in K splits completely in K[x].

My question is how they have defined splitting field here. The book up to now always defined it as a field that is associated with a specific polynomial , but this problem doesn't seem to. How can I just prove that K is a splitting field? Over which polynomial should it be a splitting field?

This definition might be useful.

"If K is an algebraic extension of F which is the splitting field over F for a collection of polynomials , then K is called a splitting field over F." (Dummit p537)