Definition 1: subfield
A set B is a subfield of a set A if both A,B are fields and B is a subset of A.
Definition 2: field extension
A set A is a field extension of a set B if B is a subfield of A.
Definition 3: R=F(u)
A field R = F(u) if and only if the following is true:
(i) u is a number that may or may not be part of the field F
(ii) For any numbers x ∈ R, there exists coefficients a0, ..., an such that all ai ∈ F and x = a0un + a1un-1 + ... + an.
The important idea behind Definittion 3 above is that F(u) is a field extension of F.
- Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001