1) Why do we have to specifically use only irreducible polynomials to

construct fields? and are we obtain irreducible polynomial let say for the field of $\displaystyle GF(2^2) $?

2) Why primitive elements/polynomials are important?

3) What does it means by conjugates. Here is a statement saying that

neither /alpha nor its conjugates, /alpha^j are elements to which /beta is

mapped.