Automorphisms of Finite Fields

  • Thread starter isaiah
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  • #1
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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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Answers and Replies

  • #2
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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?
 
  • #3
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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.
 
  • #4
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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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