How can I construct $\Bbb{Q}$-algebras of dimension $4$.
Some examples that comes to mind are $\Bbb{Q} \oplus \Bbb{Q} \oplus \Bbb{Q} \oplus \Bbb{Q}$ , rational quaternions $\Bbb{H}$ and $M_2(\Bbb{Q})$ ?
Is there a way to determine all of them upto isomorphism? How many are there?
$\Bbb{Q}$-algebras of dimension $< 4$ must be commutative, right?
So they will be $\Bbb{Q} \oplus \Bbb{Q} \oplus \Bbb{Q}$ and $\Bbb{Q} \oplus \Bbb{Q} $ only.
Similarly how to go about finding , say $\Bbb{Q}$-algebras of dimension $n$.