I have a topological field K that admits a non-trivial continuous exponential function E, must every non-trivial continuous exponential function E′ on K be of the form E ′ ( x ) = E ( r σ ( x ) ) for some r ∈ K* and σ ∈ A u t ( K / Q )?If not, can you explain for which fields other than R is this condition met?

Question

I have a topological field K that admits a non-trivial continuous exponential function E, must every non-trivial continuous exponential function E′ on K be of the form       E    ′    (  x  )  =  E  (  r  σ  (  x  )  ) for some   r  ∈  K* and   σ  ∈  A  u  t  (  K      /        Q    )?If not, can you explain for which fields other than       R   is this condition met?

Layne Murillo · Accepted Answer

It seems that as-stated, the answer is false. I&#039;m not satisfied with the following counterexample, however, and I&#039;ll explain afterwards.Take   K  =      C   and let   E  (  z  )  =      e    z   be the standard complex exponential. Take       E    ′    (  z  )  =            e      z        ¯    =      e                  z        ¯            , where       z    ¯   is the complex conjugate of z. Then E′(z) is not of the form E(rz), and yet is a perfectly fine homomorphism from the additive to the multiplicative groups of       C  .Here&#039;s why I&#039;m not satisfied: you can take any automorphism of a field and cook up new exponentials by post-composition or pre-composition. In the case I mentioned, these two coincide.

Antwan Perez · Accepted Answer

This won&#039;t work in       R   because there are no nontrivial continuous automorphisms there. It would be interested in seeing an answer to a reformulation to this problem that reflected this.

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New Questions in Calculus 1