Root-embedding correspondence (${\mathrm{Root}}_f(L)\cong {\mathrm{Hom}}_K(K[t]/(f),L)$)

A root of an irreducible polynomial in an extension field defines a base-field embedding of the polynomial quotient by evaluation. Conversely the image of the residue class of the variable determines the embedding and is a root. This counts embeddings through separable minimal polynomials.