I know that there is a division algebra over $\mathbb{Q}$ such that it is algebraic and infinite dimensional over it's center i.e. $\mathbb{Q}$. But for construct this division algebra. we can use prime numbers.

I want to know a general method of construction for every arbitrary field.

I know that there is a division algebra over $\mathbb{Q}$ such that it is algebraic and infinite dimensional over it's center i.e. $\mathbb{Q}$. But for construct this division algebra. we can use prime numbers.

I want to know a general method of construction for every arbitrary field. Actually, for an arbitrary field $F$, I want to build a division ring like $D$, such that $Z(D)=F$ and $[D:F]=\infty$ and $D$ be algebraic over $F$ (i.e. every element of $D$ be algebraic over $F$). For example by Hilbert's method for constructing division rings (from fields that have a non-trivial automorphism), we can make some division rings that have $Z(D)=F$ and $[D:F]=\infty$ but $D$ isn't algebraic over $F$. So I want to build division ring like Hilbert but be algebraic.