# Thread: Doubt in Galois Field

1. ## Doubt in Galois Field

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 $GF(2^2)$?

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.

2. Originally Posted by classic_phohe
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 $GF(2^2)$?
Because if $f(x)$ is not irreducible then $\text{GF}(p)/f(x)$ is not necessarily a field!
And so it cannot be an extension field.

2) Why primitive elements/polynomials are important?
Here is one example. Say $K/F$ is primitive, then it means $K=F(\alpha)$ for some $\alpha \in K$. If you want to compute $\text{Gal}(K/F)$ all you need to do is compute the irreducible polynomial for $\alpha$ over $F$ and define the automorphism $\alpha \to \alpha_i$ where $\alpha_i$ is a root of this polynomial in $K$. Therefore, an easy way to see that $K/F$ is Galois is if and only if $K$ contains all the zeros of the minimal polynomial of $\alpha$ with not repeated roots.

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.
Let $K/F$ be an algebraic extension.
Let $\alpha,\beta \in K$. We say that $\alpha,\beta$ are conjugates iff they have the same minimal polynomials over $F$.