Automorphisms of Finite Fields

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

Say for example $$F_{2^{29}}$$, $$F_{2^{58}}$$ and $$F_{2^{116}}$$

Let $$\alpha \in F_{2^{58}}$$\$$F_{2^{29}}$$

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

If not why not since the orbit of $$\alpha$$ under this automorphism will be 58.

Does it mean that the other elements shift to $$F_{2^{116}}$$?

Isaiah.

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