Exercise 12, § 14

Let K be a commutative field, complete for a discrete valuation $v($Comm. Alg., VI) that we assume normalized (that is, $v(K^*) =\mathbf{Z}$). Denote the ring of $v$ by A and its residue field by $\kappa$. Recall (loc. cit.) that if L is a finite extension of K, then the valuation $v$ admits a unique extension $v_L$ to L; we say that the extension is unramified if $v_L$ has integer values.

a) Let D be a field with center K; let Nrd$: D\rightarrow K$ be the reduced norm on D (VIII, p. 340, Definition 2). Prove that the mapping $w_0=v\circ$ Nrd is a valuation on D (to prove the inequality $w_0(x+y)\geqslant$ inf($w_0(x), w_0(y)$), reduce to the case $y= 1$, and observe that the restriction of $w_0$ to $K(x)$ is a multiple of $v_{K(x)}$). Denote by $w$ the normalized valuation on D equivalent to $w_0$.

b) An element $x$ of L is integral over A if and only if we have $w(x)\geqslant 0$; the set of these elements forms a subring B of D.

c) Assume from now on that the field $\kappa$ is perfect. Prove that if D is distinct from K, then it contains an unramified extension of K distinct from K. (In the opposite case, observe that we have $[\kappa_L:\kappa ] = 1$ for every extension L of K contained in D. Let $b\in B$, and let $\pi$ be a uniformizer of D. Apply the previous remark to the field $K(b)$, and prove that $b$ is in the closure of $A(\pi )$. Since the subspace $K(\pi )$ is closed in D (EVT, I, §2, n$^o3$, p. 14, corollaire 1), we obtain $B\subset K(\pi )$ and finally $D = K(\pi ).$) d) Prove that there exists a maximal commutative subfield of D that is unramified over K. (Use induction on the reduced degree of D by considering a subfield L of D that is unramified over K and applying the induction hypothesis to the commutant of L in $D.$)$*$