Let be a field, and let be an extension of of finite degree.
- Fix . Prove that the mapping ‘multiplication by ‘ is an -linear transformation on . (In fact an automorphism for .)
- Deduce that is isomorphically embedded in .
Let . Certainly then we have for all and ; so is an -linear transformation. If , then evidently .
Fix a basis for over ; this yields a ring homomorphism which takes and returns the matrix of with respect to the chosen basis. Suppose ; then for all , and thus . So is injective as desired.