# Finite Field

• Feb 9th 2009, 08:10 PM
classic_phohe
Finite Field
An irreducible polynomial P(x) of degree m over GF(q) can be used to construct an extension field of GF(q). What is actually meant by constructing an extension field?

and why is it important to ensure whether a monic polynomial of degree m over GF(2^n) is primitive???

lastly, there are 3 bases of finite field: polynomial, normal and dual base. in what condition each of the bases is favored over the others?

thank you very much for your help
• Feb 9th 2009, 08:38 PM
ThePerfectHacker
Quote:

Originally Posted by classic_phohe
An irreducible polynomial P(x) of degree m over GF(q) can be used to construct an extension field of GF(q). What is actually meant by constructing an extension field?

If $\displaystyle p(x)$ is irreducible and $\displaystyle \mathbb{F}_q = \text{GF}(q)$ then $\displaystyle \mathbb{F}_q[x]/(p(x))$ is a field. Furthermore, $\displaystyle \mathbb{F}_q$ can be identified as a subfield of this field by an embedding $\displaystyle a \mapsto a + (p(x))$. And so $\displaystyle \mathbb{F}_q[x]/(p(x))$ can be regarded as an extension field of $\displaystyle \mathbb{F}_q$. In fact it is an extension of degree $\displaystyle m$, do you see why?

And I am not sure about you other question, I do not know what you mean by 'primitive'?
• Feb 9th 2009, 08:44 PM
classic_phohe
Quote:

Originally Posted by ThePerfectHacker
If $\displaystyle p(x)$ is irreducible and $\displaystyle \mathbb{F}_q = \text{GF}(q)$ then $\displaystyle \mathbb{F}_q[x]/(p(x))$ is a field. Furthermore, $\displaystyle \mathbb{F}_q$ can be identified as a subfield of this field by an embedding $\displaystyle a \mapsto a + (p(x))$. And so $\displaystyle \mathbb{F}_q[x]/(p(x))$ can be regarded as an extension field of $\displaystyle \mathbb{F}_q$. In fact it is an extension of degree $\displaystyle m$, do you see why?

And I am not sure about you other question, I do not know what you mean by 'primitive'?

I dun quite get it yet.

Im not quite sure what is primitive but what I read is that a monic polynomial of degree m with maximum order is said to be primitive
• Feb 9th 2009, 08:59 PM
ThePerfectHacker
Quote:

Originally Posted by classic_phohe
I dun quite get it yet.

What does not make sense? Have you ever talked about talking a ring modulo an ideal in class? This is precisely what is being down. The ideal here is $\displaystyle (p(x))$ (or in notation $\displaystyle \left< p(x)\right>$ is you perfer) and the ring here is $\displaystyle \mathbb{F}_q[x]$. I am asking if you form this factor ring you get a field, and this field is an extension field of $\displaystyle \mathbb{F}_q$. By this method you can construct extension fields if you are given an irreducible polynomial.
• Feb 9th 2009, 09:08 PM
classic_phohe
Quote:

Originally Posted by ThePerfectHacker
What does not make sense? Have you ever talked about talking a ring modulo an ideal in class? This is precisely what is being down. The ideal here is $\displaystyle (p(x))$ (or in notation $\displaystyle \left< p(x)\right>$ is you perfer) and the ring here is $\displaystyle \mathbb{F}_q[x]$. I am asking if you form this factor ring you get a field, and this field is an extension field of $\displaystyle \mathbb{F}_q$. By this method you can construct extension fields if you are given an irreducible polynomial.

erm..nope..I didnt take this subject in class, Im reading this on my own interest. So I really found a lot of doubt in it. what about composite field? what is the purpose of having composite field?