-- Some questions about the definition of a splitting field of a polynomia

Howdy,


Given a polynomial f(x) in F[x], where F is a field,
one defines a splitting field of f(x) as an extension K of F containing the roots of f; and that K is also generated by the roots of f over F, i.e.
K = F(a_1, .., a_n), where the a_i are the roots of f(x).

Fair enough, but what *is* F(a_1, .., a_n)? Rather, where does *it* live?
It seems like this definition is just assigning a letter K to the object F(a_1, .., a_n) and calling it a splitting field of f (and, of course, it doesn't make much sense to talk about the subfield generated by a subset, without reference to an ambient field containing that subset).


K itself must be (?) the subfield of *some field* generated F and the roots of f, yes?

The subfield of what, though? The algebraic closure of F?
This definition seems roundabout to me: it's basically telling me that the splitting field is F(a_1, .., a_n), and leaves it at that.

When I think of the splitting field of x^2 - 2 over Q, for instance, I know that the roots live in R; and then I can generate these roots over Q to get the splitting of x^2 - 2 over Q.

But what's wrong with taking the subfield of the algebraic closure Q* of Q generated by the roots of
x^2 - 2 over Q and calling *it* the splitting field of x^2 - 2 over Q?

Are these generated subfields one and the same? Why?


So what's going on? When thinking about the splitting field of some polynomial f(x) in F[x], should I first think of an algebraic closure F* of F
(since it contains all the roots of f anyway), and then form the subfield of F* generated by the roots of f over F, and call it a splitting field of f over F?



I hope to have been clear in spite of my confusion!

I look forward to clarification. Thank you!
.

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