I want to understand about existence of some non-commutative division algebras over $\mathbb{Q}$ of dimension $9$.
Q. Does there exist a division algebra $D$ such that
$D$ is non-commutative;
$D$ is of dimension $9$ over $\mathbb{Q}$ and $Z(D)=\mathbb{Q}$;
$K:=\mathbb{Q}(2^{1/3})$ is a maximal sub-field of $D$?
My way towards solution: if $D$ is such algebra, then consider $x\in D$ outside $K:=\mathbb{Q}(2^{1/3})$. If conjugation by $x$ leaves $K$ invariant then it induces an automorphism of $K$ which fixed $\mathbb{Q}$; the only possibility of this is trivial automorphism, which means $x$ centralizes $K$, contradiction.
In fact, we can show that $D=K\oplus xK \oplus x^2K$ as a vector space.
Further $x^3$ centralize all generators of $D$, so $x^3\in\mathbb{Q}\setminus \{0\}$. Next, how should I proceed to determine structure of $D$?
I never studied division algebras other than quaternions and fields. I do not know how this question will be, but I was trying to see whether after $2^2$, can we get non-commutative division algebra whose dimension over its center is $3^2$? So a simple case I thought is through above questions, I was unable to complete the solution of existence.