Definition 1

Let A be a local ring. A field of representatives of A is any subfield K of A such that the canonical homomorphism of A onto $\kappa_A$ induces an isomorphism of K onto $\kappa_A$ (in other words, such that $A = K + m_A$).

There can exist a field of representatives of A only if A has a characteristic. This condition is sufficient when A is separated and complete. More precisely, one has the following theorem: