Lemma 8.3 (Kummer’s Lemma). Assuming that:

• $(K,\left|\bullet \right|)$ a complete discretely valued field

• $f(X)\in K[X]$ a separable irreducible polynomial with roots ${\alpha }_1,\dots ,{\alpha }_n\in K^{\text{sep}}$ ($K^{\text{sep}}$ is the separable closure of $K$)

• $\beta \in K^{\text{sep}}$ with

  $$|\beta -{\alpha }_1|<|\beta -{\alpha }_i|$$

  for $i=2,\dots ,n$.