I have been reading 'The Arithmetic of Elliptic Curves' by Silverman.

It proves that any elliptic curve defined over a perfect field can be reduced to Weierstrass form, [III.3.1,page 59]. The convention $k$ is perfect is mentioned at the begining of the book.

The proof uses the fact that $\bar{k}/k$ is Galois when $k$ is perfect.

Hence I don't know how to move the proof to the case $k$ is imperfect.



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