Infinite field which is not a metric space

Question

Are there any infinite fields which are not metric spaces (other than the discrete topology)?

If so, why must a finite-dimensional vector space over this field be locally compact?

Answer 1

Every non empty set can be a metric space. Just define $$d(x,y)=\left\{ \begin{array}{lcl} 0&\text{ if }&x=y\\ 1&\text{ if }&x\neq y \end{array} \right.$$

Answer 2

Any discrete space is locally compact, and finite dimensional vector spaces over a field with the discrete topology also has the discrete topology.

I think what you want to consider are metrics which induces a topology with respect to which the field operations are continuous.

I'm no expert on this, but you might be interested in http://en.wikipedia.org/wiki/Local_field

I believe fields like $\mathbb{F}_p(t)$ and its finite extensions are examples of infinite fields which do not admit a metric w.r.t. to which the field operations are continuous.