Neukirch's book defines a global field as a finite extension of $\mathbb Q$ or a finite extension of $\mathbb{F}_p(x)$ for $p$ prime, and a local field as a field that admits some discrete valuation $v$ with which $K$ is complete and has finite residue class field. Neukirch then states that the completion of a global field is always a local field.
However, I am having to difficulties:
I don't know how to prove that every valuation of a global field is discrete;
I can prove that a completion of a global field has finite residue class field provided the following fact (that I can't prove) is true:
"Let $L/K$ be a finite extension. Then $\hat{L}/\hat{K}$ is also a finite extension, where $\hat{L}$ and $\hat{K}$ is the completion of the fields $L$ and $K$ with respect to some valuation $v\colon L\to \mathbb R\cup\{\infty\}$".