Editing Tensor product of fields (section)

{{Short description|Ring produced from two fields}}{{Other uses|Tensor product (disambiguation)}}{{distinguish|Tensor field}}
{{textbook|date=August 2021}}

In [[mathematics]], the '''tensor product''' of two [[field (mathematics)|fields]]  is their [[tensor product of algebras|tensor product]] as [[algebra over a field|algebras]] over a common [[subfield (mathematics)|subfield]]. If no subfield is explicitly specified, the two fields must have the same [[characteristic (algebra)|characteristic]] and the common subfield is their [[prime field|prime subfield]].

The tensor product of two fields is sometimes a field, and often a [[product ring|direct product]] of fields; In some cases, it can contain non-zero [[nilpotent element]]s.

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