I am going through a paper, https://www.sciencedirect.com/science/article/abs/pii/0196677488900144 and over there I found this statement in pg 2 confusing-
The structure of associative algebra over $\mathbb{Q}$ is of the form $M_n(F)$, where $F$ is a field containing $\mathbb{Q}$.
I looked up wikipedia https://en.wikipedia.org/wiki/Quaternion_algebra and there I found,
Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension $K$ of $F$, ${\displaystyle A\otimes _{F}K}$ is isomorphic to the 2×2 matrix algebra over K. But it did not become clear.
Another point which stuck from the paper and also in wikipedia it is written the same.
A quaternion algebra $(a,b)_F$ is either a division algebra or isomorphic to the matrix algebra of $2×2$ matrices over $F$.
The above statement follows from Wedderburn Artin structure theorem according to the paper. But on looking into that, the thing did not become clear.
Can anyone help me out understanding those two points I mentioned, or maybe suggest some materials to read?