An example of noncommutative division algebra over $Q$ other than quaternion algebras

Question

Could anyone please show me an example of finite dimensional noncommutative associative division algebra over the field of rational numbers $Q$ other than quaternion algebras?

Answer 1

There a many constructions. A simple one is the following:

Let $K/F$ be a cyclic extension of fields of degree $n$ and let $\sigma$ be a generator of the Galois group $G=Gal(K/F)$. Let $N_{K/F}(K^\times)$ be the subgroup of the multiplicative group $F^\times$ of norms of non-zero elements of $K$, and let $\alpha\in F^\times$ be an element of order $n$ modulo $N_{K/F}(K^\times)$. Then the $K$-vector space $$A=K\oplus xK\oplus x^2K\oplus\cdots\oplus x^{n-1}K$$ with basis $\{1,x,\dots,x^{n-1}\}$ is an $F$-algebra with multiplication extending that of $K$ and such that $$kx=x\sigma(k), \qquad\forall k\in K$$ and $$x^n=\alpha$$ is an associative division $F$-algebra.

So we can do this general contruction with $F=\mathbb Q$. For example, the polynomial $f=X^3-3X-1\in\mathbb Q[X]$ has discriminant $81$, which is a square, so its Galois group $G$ has only even permutations: since $f$ is irreducible, we see that $G$ is cyclic of order $3$.

Let $\zeta$ be a root of $f$ in $\mathbb Z$, and let $K=\mathbb Q[\zeta]$. Then $K/\mathbb Q$ is a cyclic extension of degree $3$. A generator $\sigma$ of the Galois group maps $\zeta$ to $2-\zeta^2$. The norm of the element $a+b\zeta+c\zeta^2$ is $$N_{K/F}(a+b\zeta+c\zeta^2)=a^3+6 a^2 c-3 a b^2-3 a b c+9 a c^2+b^3-3 b c^2+c^3.$$ Now we need an number $\alpha\in\mathbb Q$ such that its cube is in the image on $N_{K/\mathbb Q}$ but which itself isn't there. I think $2$ works (its cube is $N(-2\zeta-2\zeta^2)$, but one has to check that $2$ is not a norm...) The above construction, then, provides us with a rational division algebra of dimension $9$.

Can someone check that $2$ is not a norm?