Extension Field

Just like how we create complex numbers from real numbers by adjoining a root of \(x^{2} + 1 = 0\) whose root we call as \(i\), we create extension fields by adjoining roots of irreducible polynomials who have no solutions in the base field.

Steps:

1. Start with a base field \(F\)
2. Find an irreducible polynomial \(f(x)\) over \(F\)
3. Adjoin a root \(\alpha\) by declaring \(f(\alpha) = 0\)
4. The extension field consists of all polynomials in \(\alpha\) with coefficients from \(F\) where \(f(\alpha) = 0\) is used to reduce the powers.
5. The extension field has \(+\) and \(*\) operations where the coefficients modulo order of \(F\) and the polynomials module \(f(x)\)