On skew field coproducts with a finite extension

It is shown that the field coproduct of any skew fieldE with a binomial (commutative) field extensionF/k overk can be expressed as a cyclic extension of a skew fieldK (theE-socle), itself the field coproduct of F:k] copies ofE overk. Qua vector space the coproduct may also be expressed as a tensor product ofE andK overk. To the memory of Shimshon Amitsur