The theory of Severi-Brauer varieties is well-known. Let $k$ be a (perfect) field. There may exist varieties not isomorphic to $\mathbf{P}^n$ over $k$, which are isomorphic to $\mathbf{P}^n$ over $\overline{k}$. They are classified by $H^1(k,\mathrm{PGL}\_n)$.


How about quadrics? Say $k$ is a (perfect) field and $X$ is a smooth, projective $k$-variety of dimension $n$. Assume that $X \otimes\_k \overline{k}$ is isomorphic to a quadric. Is $X$ necessarily an $n$-dimensional quadric itself? If not, can you give some nice examples (e.g. over number fields) which show that this need not be the case?



"Forms" of quadrics
It is not too difficult to see that any automorphism of a smooth quadric hypersurface
$$X : Q(x) = 0,$$ over a field $k$ must be a projective automorphism (see for instance the argument I give in [Automorphism group of a smooth quadric $Q\subset\mathbb{P}^4$](https://mathoverflow.net/questions/178932/automorphism-group-of-a-smooth-quadric-q-subset-mathbbp4/178938#comment448489_178938)). Hence the automorphism group of any quadric $X$ is the projective orthogonal group $\mbox{PO}(Q)$. Thus twists of $X$ are parametrised by $H^1(k, \mbox{PO}(Q))$.


As to your question: yes there exist twists of quadrics which are not themselves quadrics.


Recall that $\mathbb{P}^1 \times\mathbb{P}^1$ embeds into $\mathbb{P}^3$ via the Segra embedding as a quadric surface. So let now $k$ be a field for which there exists two conics $C\_1$ and $C\_2$ over $k$ without rational points, and take


$$X = C\_1 \times C\_2.$$


This is a twist of a quadric surface, but is not isomorphic to a quadric surface. To see this, note that any quadric surface contains an effective divisor $D$ of self-intersection $D^2 = 2$. However it is not too difficult to see that for any effective divisor $D$ on $X$ we have $D^2 = 0$ or $D^2 \geq 8$, hence $X$ is not isomorphic to a quadric surface, as required.


To see what is happening in general, note that those twists of a quadric hypersurface $X$ which are themselves quadrics are parametrised by $H^1(k,\mbox{O}(Q))$. Thus, there are twists of a quadric which are not quadrics whenever the map 
$$H^1(k,\mbox{O}(Q)) \to H^1(k,\mbox{PO}(Q)),$$
is not surjective. I believe this is the case exactly for quadrics hypersurfaces in $\mathbb{P}^{2n+1}$ over a field $k$ for which the Brauer group $\mbox{Br}(k)$ of $k$ has non-trivial $2$-torsion, but I did not check all the details.


