How Do Automorphisms Affect Elements in Non-Prime Subfields of Finite Fields?

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Discussion Overview

The discussion centers around the effects of field automorphisms on elements within finite fields, specifically focusing on non-prime subfields. Participants explore the implications of automorphisms in the context of finite extensions of finite fields, particularly regarding the relationships between fields like F_{2^{29}}, F_{2^{58}}, and F_{2^{116}}.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Isaiah questions whether an element \(\alpha\) from \(F_{2^{58}} \setminus F_{2^{29}}\) can become an element of \(F_{2^{29}}\) under the action of automorphisms \(\sigma^i\) for \(1 \leq i \leq 58\), and seeks clarification on the implications of the orbit of \(\alpha\).
  • Another participant notes that a finite extension is normal if every automorphism sends the base field to itself, suggesting that any automorphism of \(F_{2^{58}}\) must send \(F_{2^{29}}\) to itself, which may address Isaiah's question.
  • Isaiah further inquires about the fate of elements in \(F_{2^{58}} \setminus F_{2^{29}}\), questioning whether they are mapped to \(F_{2^{29}}\) or remain in \(F_{2^{58}} \setminus F_{2^{29}}\).
  • Another participant asserts that since automorphisms are one-to-one and onto, if an automorphism fixes a subset \(B\), it must also fix the complement \(A \setminus B\), implying that elements not in \(F_{2^{29}}\) remain outside of it.

Areas of Agreement / Disagreement

Participants express differing views on the behavior of elements under automorphisms, particularly regarding whether elements from \(F_{2^{58}} \setminus F_{2^{29}}\) can be mapped to \(F_{2^{29}}\). The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

There are limitations in the assumptions regarding the nature of the automorphisms and the specific properties of the finite fields involved, which may affect the conclusions drawn by participants.

isaiah
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I am trying to figure out the effect of a field automorphism on a field with a non prime subfield.

Say for example [tex]F_{2^{29}}[/tex], [tex]F_{2^{58}}[/tex] and [tex]F_{2^{116}}[/tex]

Let [tex]\alpha \in F_{2^{58}}[/tex]\[tex]F_{2^{29}}[/tex]

Under [tex]{\sigma}^{i}, 1 \le i \le 58[/tex] do we get any case where [tex]\alpha[/tex] becomes an element of [tex]F_{2^{29}}[/tex] ?

If not why not since the orbit of [tex]\alpha[/tex] under this automorphism will be 58.

Does it mean that the other elements shift to [tex]F_{2^{116}}[/tex]?

Thanks in advance.

Isaiah.
 
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Remember that a finite extension E/F is normal iff every automorphism of E sends F to itself. Since any finite extension of finite fields is Galois, any automorphism of F_2^58 must send F_2^29 to itself. Does that answer your question?
 
Now what really happens to the elements of F_2^58\F_2^29 (ie exclude the ones in F_2^29)? Do they go to F_2^29 or do they remain in F_2^58\F_2^29?

Thanks,

Isaiah.
 
Well, an automorphism is one to one and onto, so if f:A->A is one to one and onto, and f(B)=B, we must have f(A\B)=A\B, mustn't we?
 

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