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Math Help - Confusion about splitting fields

  1. #1
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    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 f(x) \in F[x], 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?
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  2. #2
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    Quote Originally Posted by eeyore View Post
    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 f(x) \in F[x], 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 f(x) \in F(x), then K is called a splitting field over F." (Dummit p537)
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