Math 2.0

Field theory and polynomials Definition

Tensor product of fields

Builds on Field extensionTensor product of vector spacesCommutative ringMinimal polynomial of an algebraic element

Definition

Let KK be a field and let EE and FF be field extensions of KK. The tensor product E⊗KFE \otimes_K F is the tensor product of the KK-vector spaces EE and FF, with the unique KK-bilinear multiplication such that (a⊗b)(c⊗d)=ac⊗bdfor a,c∈E and b,d∈F.(a \otimes b)(c \otimes d) = ac \otimes bd \quad \text{for } a, c \in E \text{ and } b, d \in F. It is a commutative ring with identity 1⊗11 \otimes 1 and a KK-algebra.

Remarks

The maps a↦a⊗1a \mapsto a \otimes 1 and b↦1⊗bb \mapsto 1 \otimes b embed EE and FF into E⊗KFE \otimes_K F. If E=K(α)E = K(\alpha) is finite and ff is the minimal polynomial of α\alpha over KK, then E⊗KF≅F[x]/(f)E \otimes_K F \cong F[x]/(f). If moreover f=g1⋯grf = g_1 \cdots g_r with distinct monic irreducible gi∈F[x]g_i \in F[x], the Chinese remainder theorem gives E⊗KF≅∏i=1rF[x]/(gi)E \otimes_K F \cong \prod_{i=1}^r F[x]/(g_i), a product of fields.

Example: ℂ⊗ℝℂ≅ℂ×ℂ\C \otimes_\R \C \cong \C \times \C. Check: ℂ=ℝ(i)\C = \R(i) with f=x2+1f = x^2 + 1, and over F=ℂF = \C one has x2+1=(x−i)(x+i)x^2 + 1 = (x - i)(x + i), so ℂ⊗ℝℂ≅ℂ[x]/(x−i)×ℂ[x]/(x+i)≅ℂ×ℂ\C \otimes_\R \C \cong \C[x]/(x - i) \times \C[x]/(x + i) \cong \C \times \C.

Non-example: E⊗KFE \otimes_K F need not be a field, even when EE and FF are; ℂ⊗ℝℂ\C \otimes_\R \C has the zero divisors (1,0)(1, 0) and (0,1)(0, 1) in the product above.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptVector space, definition, no evidence levelVector spacedefinitionRing, definition, no evidence levelRingdefinitionBilinear form, definition, no evidence levelBilinear formdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionTensor product of vector spaces, definition, no evidence levelTensor product of vec…definitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionSubfield, definition, no evidence levelSubfielddefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionField extension, definition, no evidence levelField extensiondefinitionAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionMinimal polynomial of an algebraic element, definition, no evidence levelMinimal polynomial of…definitionTensor product of fields, definition, no evidence levelTensor product of fie…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Vector space definition
  3. Ring definition
  4. Bilinear form definition
  5. Commutative ring definition
  6. Polynomial ring definition
  7. Subring definition
  8. Unit of a ring definition
  9. Tensor product of vector spaces definition
  10. Degree of a polynomial definition
  11. Polynomial evaluation definition
  12. Field definition
  13. Leading coefficient definition
  14. Root of a polynomial definition
  15. Subfield definition
  16. Monic polynomial definition
  17. Field extension definition
  18. Algebraic element definition
  19. Minimal polynomial of an algebraic element definition
  20. Tensor product of fields definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Tensor_product_of_fields
About this page
TagSDJ4ADZK8M
URIm2://SDJ4ADZK8M
Kinddefinition
WritingW0 stub
MSC12F99, 15A69
DefinesTensor product of fields
UsesField extension, Tensor product of vector spaces, Commutative ring, Minimal polynomial of an algebraic element
Changed2 November 2026

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