##
Splitting polynomials and fields: Definitions and motivation

Let F be any field, and f be a monic polynomial of degree n in F[X].
This polynomial is said to __split__ in F if it factors
completely, i.e., factors as a product of n linear factors
x-r_{i}. The r_{i} are then the __roots__ of f,
that is, the solutions of the equation f(x)=0.
If K is some extension of F, we likewise say
f __splits in K__ if can be written as a product
(x-r_{1})(x-r_{2})...(x-r_{n})
of n linear factors in K[X]. Clearly f then splits also
in F(r_{1},r_{2},...,r_{n}),
the subfield of K generated by the roots.
We say that K is a __splitting field__ of f over F
if f splits in K and K=F(r_{1},r_{2},...,r_{n}).

We shall show that every polynomial has a splitting field K,
which is unique up to automorphisms. When the polynomial
is separable, the automorphisms of K that act trivially
on F (a.k.a. the ``automorphisms of K/F'')
will constitute the Galois group Gal(K/F) of K over F.

Galois himself didn't have to worry about the construction
of splitting fields, because he worked in a context in which F
is either contained in an algebraically closed field
(namely **C**), or F is finite, in which case
we shall see that F(r_{1}) is already a splitting field
because the other roots are
r^{q}, r^{q2}, r^{q3},
..., r^{qn-1}, where q=|F|.
In general we can construct a splitting field
by adjoining roots one at a time, but then we have to prove
that the resulting field does not depend on the order
in which we adjoined the roots.