# 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 $\displaystyle GF(2^2)$?

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.

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 $\displaystyle GF(2^2)$?
Because if $\displaystyle f(x)$ is not irreducible then $\displaystyle \text{GF}(p)/f(x)$ 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 $\displaystyle K/F$ is primitive, then it means $\displaystyle K=F(\alpha)$ for some $\displaystyle \alpha \in K$. If you want to compute $\displaystyle \text{Gal}(K/F)$ all you need to do is compute the irreducible polynomial for $\displaystyle \alpha$ over $\displaystyle F$ and define the automorphism $\displaystyle \alpha \to \alpha_i$ where $\displaystyle \alpha_i$ is a root of this polynomial in $\displaystyle K$. Therefore, an easy way to see that $\displaystyle K/F$ is Galois is if and only if $\displaystyle K$ contains all the zeros of the minimal polynomial of $\displaystyle \alpha$ 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 $\displaystyle K/F$ be an algebraic extension.
Let $\displaystyle \alpha,\beta \in K$. We say that $\displaystyle \alpha,\beta$ are conjugates iff they have the same minimal polynomials over $\displaystyle F$.