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 .
I can do this question which is worth 8 marks...but cannot do:
Show that if is an element of E - F2 then every non-zero element of E is a power of .
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 would like to give a detailed explanation on why the question is wrong.
As a side note, I would like to add that if , the what you said would have been true. Since then and 7 is prime.
Here(that is to say ), we have 16 elements and . Observe that this means .
Now your questions says if , then
I claim this is wrong by a counterexample:
Consider , now:
So clearly in this entire cycle never appears. And these are exactly all the powers of .
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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