Can the K-theory of a space be a field?

Question

If $X$ is a compact Hausdorff topological space, is it possible to $K(X)$ be a field considering the operations over vector bundles, $\oplus$ and $\otimes$? It is known that $K(X)$ has a ring structure, but I don't know if there exist cases when $K(X)$ is a field too.

Answer 1

The ring $K(X)$ is never a field. If $X$ is empty, then $K(X)$ is the zero ring, which is not a field. If $X$ is nonempty, choose a point $x\in X$. Then there is a restriction homomorphism $r:K(X)\to K(\{x\})=\mathbb{Z}$. But there is no field that has a ring-homomorphism to $\mathbb{Z}$: if $p$ is any prime which is not equal to the characteristic of a field $k$, then a homomorphism $k\to\mathbb{Z}$ would have nowhere to send the element $1/p\in k$.