Field isomorphism

A field isomorphism is a bijective field homomorphism; its inverse is also a field homomorphism. For fields extending a specified field $K$, an isomorphism may be required to fix $K$ pointwise, a stronger requirement than an abstract field isomorphism.

A K-isomorphism between extensions of $K$ is a field isomorphism fixing $K$ pointwise. Simple algebraic extensions $K(\alpha )$ and $K(\beta )$ admit one sending $\alpha$ to $\beta$ exactly when their minimal polynomials agree: both then identify with the same quotient $K[X]/(m)$.