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, which cannot be the whole field , because of the condition $1=\backslash sigma(1)=1$. Furthermore, any field has as ideals only the zero ideal and the whole field itself (because if there is any non-zero field element in an ideal, it is invertible, showing the ideal is the whole field). Therefore, the kernel is , so any embedding of fields is a monomorphism. Hence, is isomorphic to the subfield of . This justifies the name embedding

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