(Severi--Brauer variety)
Let \(k\) be a field with separable closure \(k^s\). A Severi–Brauer variety of dimension \(n\) over \(k\) is a variety \(X\) over \(k\) with \(X_{k^s} \cong \PP^n_{k^s}\).
Isomorphism classes of Severi–Brauer varieties of dimension \(n\) over \(k\) are in bijection with \(H^1(\Gal(k^s/k), \PGL_{n+1}(k^s))\), hence with isomorphism classes of central simple \(k\)-algebras of degree \(n+1\); each has a class in \(\operatorname{Br}(k)\).
(Châtelet) \(X \cong \PP^n_k\) if and only if \(X(k) \neq \emptyset\), if and only if its Brauer class is trivial.
The Severi–Brauer varieties of dimension \(1\) are the smooth conics. The conic \(x^2 + y^2 + z^2 = 0\) over \(\RR\) has no real point, so it is not \(\PP^1_\RR\); it corresponds to the Hamilton quaternions. If \(k\) is finite, or is the function field of a curve over an algebraically closed field, then \(\operatorname{Br}(k) = 0\) (Wedderburn, Tsen), so every Severi–Brauer variety over \(k\) is a projective space.
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