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Splitting polynomials and fields: Definitions

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}).