Exercise 2, § 8

Let $K_0$ be the field obtained by adjoining to the 2-adic field $\mathbf{Q}_2$ the roots of all the polynomials $X^{2^n} - 2$; let $v$ be the unique valuation on $K_0$ extending the 2-adic valuation and let $K$ be the completion of $K_0$ with respect to $v$. Show that the polynomial $X^2 - 3$ is irreducible in $K[X]$; let $K'$ be the field of roots of this polynomial and let $v'$ be the extension of $v$ to $K'$; then