# Thread: non-zero elements in E

1. ## non-zero elements in E

$g = X^4 + X + 1$ over F2 and let E be the extension of F2 with a root alpha of g.

Show that every non-zero element of E is a power of $\alpha$.
I can do this question which is worth 8 marks...but cannot do:
Show that if $\beta$ is an element of E - F2 then every non-zero element of E is a power of $\beta$.

Which is worth 4 marks...I'm guessing this answer is much easier since it is only 4 marks, so there is a way I can adapt my answer to the former question for this one? Or is it a totally different method?

Thanks in advance!

2. Originally Posted by hunkydory19
$g = X^4 + X + 1$ over F2 and let E be the extension of F2 with a root alpha of g.

Show that every non-zero element of E is a power of $\alpha$.
I can do this question which is worth 8 marks...but cannot do:
Show that if $\beta$ is an element of E - F2 then every non-zero element of E is a power of $\beta$.

Which is worth 4 marks...I'm guessing this answer is much easier since it is only 4 marks, so there is a way I can adapt my answer to the former question for this one? Or is it a totally different method?

Thanks in advance!
This is a question about your question. Simple curiosity. What is the definition of the field (I presume) F2?

-Dan

3. Originally Posted by hunkydory19
Show that if $\beta$ is an element of E - F2 then every non-zero element of E is a power of $\beta$.

Which is worth 4 marks...I'm guessing this answer is much easier since it is only 4 marks, so there is a way I can adapt my answer to the former question for this one? Or is it a totally different method?

Thanks in advance!
I wonder whether I have understood this correctly.

$E - F_2$ has all the elements of E except 0 and 1. Consider $\beta = \alpha^3 \in E- F_2$, We can never write $\alpha$ as a power of this $\beta$. So the statement is not true.

Either I have misunderstood the question or you have not copied it correctly

4. Originally Posted by hunkydory19
$g = X^4 + X + 1$ over F2 and let E be the extension of F2 with a root alpha of g.

Show that every non-zero element of E is a power of $\alpha$.
If $E$ is an extension field containing $\alpha$ over $\mathbb{F}_2$ then $E$ contains $\mathbb{F}_2$.
It means if $x\in \mathbb{F}_2(\alpha)$ then $x=a+b\alpha+c\alpha^2+d\alpha^3$ where $a,b,c\in \mathbb{F}_2$.
So maybe you meant to say $E=\mathbb{F}_2$.

5. I would like to give a detailed explanation on why the question is wrong.

As a side note, I would like to add that if $g(x) = x^3 + x + 1$, the what you said would have been true. Since then $\alpha^7 = 1$ and 7 is prime.

Here(that is to say $g(x) = x^4 + x + 1$), we have 16 elements and $\alpha^{15} = 1$. Observe that this means $(\alpha^3)^5 = 1$.

Now your questions says if $\beta \in E - \mathbb{F}_2$, then $\forall x \in E - \mathbb{F}_2, \exists i \in \mathbb{Z}: x = \beta^i$

I claim this is wrong by a counterexample:

Consider $\beta = \alpha^3$, now:

$\beta = \alpha^3 , \beta^2 = \alpha^6, \beta^3 = \alpha^9,\beta^4 = \alpha^{12},\beta^5 = \alpha^{15} = 1$

So clearly in this entire cycle $\alpha$ never appears. And these are exactly all the powers of $\beta$.

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Finally, I think that the question really asked you to disprove their statement.Or g(x) was different.

Hope you understand why it is wrong

6. I've attached the question to prove it's definitely minus! It comes up every year with either $g = X^3 + X + 1$ or $g = X^4 + X + 1$.

Think I'll show my lecturer your explanation Iso and see what he says!