Rational points on conics over fields of dimension 1

Question

Let $K$ be a field of cohomological dimension 1 and $C$ be a smooth projective conic over $K$. Is it true that $C$ always has a $K$-point?

Answer 1

Yes. Every such conic is a Severi-Brauer variety, so is classified by an element of the Brauer group $H^2(K, \mathbb{G}_m)$, which vanishes by hypothesis. The trivial element of the Brauer group corresponds to the Severi-Brauer variety $\mathbb{P}^1$, which in turn corresponds to the original Severi-Brauer variety having a rational point.

Note that if $K$ is a finite field you can argue more directly using the Chevalley-Warning theorem.