structural:summarization, structural:expansion, structural:reformatting
Tensor product of algebras

Let R be a commutative ring. The tensor product of R -modules applies, in particular, if A and B are R -algebras. In this case, the tensor product A ⊗ R B {\displaystyle A\otimes _{R}B} is an R -algebra itself by putting: (a 1 ⊗ b 1) ⋅ (a 2 ⊗ b 2) = (a 1 ⋅ a 2) ⊗ (b 1 ⋅ b 2). {\displaystyle (a_{1}\otimes b_{1})\cdot (a_{2}\otimes b_{2})=(a_{1}\cdot a_{2})\otimes (b_{1}\cdot b_{2}).} For example: R ⊗ R R ≅ R. {\displaystyle R\otimes _{R}R\cong R.}

A particular example is when A and B are fields containing a common subfield R. The tensor product of fields is closely related to Galois theory: if, say, A = R / f (x), where f is some irreducible polynomial with coefficients in R, the tensor product can be calculated as: A ⊗ R B ≅ B / f (x) {\displaystyle A\otimes _{R}B\cong B/f(x)} where now f is interpreted as the same polynomial, but with its coefficients regarded as elements of B. In the larger field B, the polynomial may become reducible, which brings in Galois theory. For example, if A = B is a Galois extension of R, then: A ⊗ R A ≅ A / f (x) {\displaystyle A\otimes _{R}A\cong A/f(x)} is isomorphic (as an A -algebra) to the A deg (f) {\displaystyle A^{\operatorname {deg} (f)}}.