Artin in his book, Geometric Algebra, says the connection between the left degree and right degree of a skew-field extension is unknown. Since I'm not an expert, I was wondering if someone knew the answer to this question. The book is rather old and there must have been some developments since that time.
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Anything that might happen does happen.
In
Schofield, A. H., Artin’s problem for skew field extensions, Math. Proc. Camb. Philos. Soc. 97, 1-6 (1985). ZBL0574.16008.
it is shown that for any integers $m,n>1$ there is a skew field extension with left degree $m$ and right degree $n$.
Cohn had previously proved the corresponding fact for an arbitrary pair of cardinals (greater than $1$) under the assumption that at least one is infinite.
answered Jan 10, 2019 at 9:39