Proposition 6

Let A be a complete regular noetherian local ring, whose residue field $\kappa_A$ has characteristic 0. Put $n = \dim(A)$. Then A is isomorphic to the ring of formal power series $\kappa_A[[X_1, ..., X_n]]$.

B) $p \neq 0$. A Cohen subring of A is any subring V of A which is a $p$-ring such that $A = m_A + V$ (IX, § 2, no 2, def. 2). The ring V is local; its maximal ideal $m_V$ is generated by $p.1_V$; hence one has $m_A \cap V = m_V$ and the canonical injection of V into A defines by passage to the quotient an isomorphism of the field $\kappa_V$ onto the field $\kappa_A$. If $p.1_A = 0$, V is a field of characteristic $p$. Otherwise V is a discrete valuation ring whose field of fractions has characteristic zero (IX, § 2, no 1, cor. 1 to prop. 1). One proves (IX, no 2, th. 1) that A possesses Cohen subrings.