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there is for example field $GF(2^4)=GF(16)$. Is $GF(16)$ a subfield of itself? Following this definition there is nothing written that subfield must contain less elements than original field.

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Yes, a field is (trivially) a subfield of itself. If we want to clearly state that something is a subfield of F and not F itself, we might say "proper subfield".

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