Editing Embedding (section)

===Field theory===

In [[field theory (mathematics)|field theory]], an '''embedding''' of a [[field (mathematics)|field]] <math>E</math> in a field <math>F</math> is a [[ring homomorphism]] {{nowrap|<math>\sigma:E\rightarrow F</math>}}.

The [[Kernel (algebra)|kernel]] of <math>\sigma</math> is an [[ideal (ring theory)|ideal]] of <math>E</math>, which cannot be the whole field <math>E</math>, because of the condition {{nowrap|<math>1=\sigma(1)=1</math>}}. 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 <math>0</math>, so any embedding of fields is a [[monomorphism]]. Hence, <math>E</math> is [[isomorphic]] to the [[Field extension|subfield]] <math>\sigma(E)</math> of <math>F</math>. This justifies the name ''embedding'' for an arbitrary homomorphism of fields.