Field Extension

The content in today's blog, assumes that you feel comfortable with the idea of fields. For review of this concept, start here.

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, ..., a_n such that all a_i ∈ F and x = a₀uⁿ + a₁u^n-1 + ... + a_n.

The important idea behind Definittion 3 above is that F(u) is a field extension of F.

References

• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001