I am currently reading "Algebraic Number Theory" by Neukirch and I am a bit confused by what is here. It is on page 143, it states this:
$(K,v)$ is a nonarchimedian valued field and $(\hat{K},\hat{v})$ is it's completion. Define $K_{v}$ as the separable closure of $K$ in $\hat{K}$. He says
"$K_{v}$ will not, as a rule, be complete."
I don't know what he means by this. Is this some assumption that we make? Even in the case when char$(K)=0$ and the separable closure and algebraic closure are the same this doesn't seem obvious like he says. Any information would be great. Thanks.
The field $K_v$ is an intermediate field between the fields $K$ and $\hat{K}$. By the definition of "completion," $K_v$ will only be complete if it is actually equal to $\hat{K}$.
Here's an example in the archimedean case: Consider the field of rational numbers $\mathbb{Q}$ together with the ordinary absolute value. Its completion is the field of real numbers $\mathbb{R}$. But $\mathbb{R}$ contains elements like $\pi$ that are transcendental over $\mathbb{Q}$, so the separable closure of $\mathbb{Q}$ in $\mathbb{R}$ must be a strict subfield of $\mathbb{R}$, and thus cannot be complete.