Corollary

Tag 0CPZ, permanent. Cite it as [Bourbaki, Tag 0CPZ].

Commutative Algebra, chapter VIII, § 5. Anneaux locaux réguliers

Suppose that A_p is a discrete valuation ring. Then B_q is a discrete valuation ring, t is a uniformizer of B_q, and one has

f(B_q/A_p) = 1 , \quad e(B_q/A_p) = [L : K]

(VI, § 8, n° 1).

In fact, discrete valuation rings are regular noetherian local rings of dimension 1; putting d = [L : K], one has t^d \in m_A - m_A^2, whence d = e(B_q/A_p). One has [\kappa_B : \kappa_A] = 1, whence f(B_q/A_p) = 1.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.