In which dimensions does a division algebra over a given ground field exist? (Q865265)

Let, for a field \(k\), \(N(k)\) denote the set of all natural numbers \(n\) such that a division algebra (not necessarily associative) of dimension \(n\) over \(k\) exists. It is proved that if \(k\) is algebraically closed, \(N(k)=\{1\}\). It is shown that if \(k\) is real closed, then \(N(k)=\{1,2,4,8\}\) and \(N(k)\) is unbounded if \(k\) is neither algebraically closed nor real closed.