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.
