Let $\mathbb{Q}_{p}$ be the $p$-adic field, $\overline{\mathbb{Q}_{p}}$ be its algebraic closure and $\mathbb{C}_{p}$ the completion of the latter. There is a unique well defined norm $|\cdot|$ on $\mathbb{C}_{p}$ that extends the $p$-adic norm on $\mathbb{Q}_{p}$. I am interested in the possible values of this norm.
It is obvious that the possible values of this norm on $\overline{\mathbb{Q}_{p}}$ are fractional powers of $p$ - this is just clear from the definition. In Neil Koblitz’s book “p-adic Numbers, p-adic Analysis, and Zeta-Functions”, the author claims that the same is true for $\overline{\mathbb{Q}_{p}}$, but leaves the proof as an exercise.
For me this is not easy to see, since the set of fractional powers of $p$ is obviously dense in $\mathbb{R}$, so a-priori it seems that any positive real number can be a possible value of the norm.
My main idea is to somehow use Krasner’s Lemma to show that if $\{x_{n}\}$ is a Cauchy-sequence in $\overline{\mathbb{Q}_{p}}$, then there must be a subsequence that is contained in a finite field extension of $\mathbb{Q}_{p}$, but I can’t seem to achieve this…
Is this fact true? Are the possible values of the norm of elements of $\mathbb{C}_{p}$ only fractional powers of $p$? How can one prove this? Any hints or references would be greatly appreciated.
The following is a standard exercise in ultrametrics: Let $K, \lvert \cdot \rvert$ be any field with an ultrametric absolute value. Use the ultrametric property to show: If $(x_n)_n$ is a Cauchy sequence in the field which does not converge to $0$, then the sequence of values $\lvert x_n \rvert$ is eventually constant. (Do this exercise yourself. For proofs on this site, cf. Limit of non-archimedean absolute values is eventually constant and If $(x_n)_{n\geq 1}$, $x_n \in \mathbb{Q}$ is p-adic Cauchy, show ord$_p(x_n)$ eventually constant. .)
Now, by definition, every element of $\mathbb C_p$ is the limit of a Cauchy sequence of elements in $\overline{\mathbb Q_p}$.