Corollary

Let A be a regular noetherian local ring and let P $\in A[T]$ be an Eisenstein polynomial for $m_A$. The ring $B = A[T]/(P)$ is a regular noetherian local ring, of the same dimension as A, and one has $[ \kappa_A : \kappa_B ] = 1$. Finally, one has $m_B = B.m_A + Bt$, where t is the class of T modulo (P).

The case where P has degree $\geq 2$ follows from prop. 4, where one takes $r = 1$; when P has degree 1, that is to say is of the form $T - c$ with $c \in m_A$, the corollary is immediate.