Two comments: 

A) There is no such thing as "different finite fields of the same size with 
the same generator name"

sage: GF(4, 'a') is GF(4, 'a')
True
sage: GF(4, 'a') is GF(4, 'b')
False

There is no way to cast finite field elements to other finite field 
elements apart from the prime subfield case.

B) The ambiguity in mapping GF(p^n) to itself is of course super-important 
and gives rise to lots of interesting mathematics.

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