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I'm trying to find all normal bases of GF(16)/GF(2) and then calculate the traces $Tr_{GF(16)/GF(2)}$ and $Tr_{GF(16)/GF(4)}$.

I'm already failing on the first part. I've read the wikipedia article about "Normal basis", where there is a short discussion of the normal basis of GF(16)/GF(2), but it didn't help me. So could anyone explain how to find the bases?

Studentu
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  • No, I've seen that post before, but thanks for your comment. That issue doesn't help me, because it discusses a "special case where a so calles optimal basis is available". And it talks about elements being conjugates of each other - I don't know what this means. Futhermore, I don't see how to use the procedure for that specific example (GF(2^60)) for my case with GF(2^4). – Studentu Dec 19 '21 at 16:22
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    @Studentu: Then please edit that into the question, because it substantially changes what you're asking. – Jacob Manaker Dec 20 '21 at 05:12
  • You say the Wikipedia article "didn't help me," which suggests that you have launched into an exercise "to find all normal bases" without understanding how to construct $GF(p^k)$ from $GF(p)$. The definition of normal basis requires an understanding of "elements being conjugates," so perhaps you should first research that and ask about anything that needs clarification. You may find the mystery of this problem then evaporates. – hardmath Dec 20 '21 at 19:18

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