Tensor product of fields

This article reads like a textbook. Please improve this article to make it neutral in tone and meet Wikipedia's quality standards. (August 2021)

In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfield.

The tensor product of two fields is sometimes a field, and often a direct product of fields; In some cases, it can contain non-zero nilpotent elements.

The tensor product of two fields expresses in a single structure the different way to embed the two fields in a common extension field.