Exercise 21, § 8

Let K be a field, $v$ a discrete valuation on K, A the ring of $v$, $\pi$ a uniformizer of $v$ and k the residue field of $v$.

(a) Let P, R be two polynomials in $A[X]$, P being monic, and suppose that: (1) $\deg(R) < h \cdot \deg(P)$, where $h \geq 1$ is an integer; (2) the canonical image $\overline{P}$ of P in $k[X]$ is irreducible. Show then that, if the polynomial $Q = P^h + \pi R$ is reducible in $\mathbf{K}[X]$, necessarily $h > 1$ and $\overline{P}$ divides the canonical image $\overline{R}$ of R in $k[X]$. Deduce that, for a given monic irreducible polynomial $r(X) \in k[X]$ and a given integer $h \geq 1$, there exists a separable irreducible polynomial $Q \in A[X]$ such that its canonical image $\overline{Q}$ is equal to $r^h$.

(b) Let $k' = k(\alpha)$ be an algebraic extension of k of degree m and $h$ an integer $\geq 1$. Show that there exists an algebraic extension L of K of degree $hm$ such that there is only a single valuation $v'$ (up to equivalence) on L extending $v$, that $e(v'/v) = h, f(v'/v) = m$ and that the residue field of $v'$ is isomorphic to $k'$ (use (a)).