Math 2.0

Field theory and polynomials Definition

Trace form

Builds on Finite extensionField traceSymmetric bilinear formNondegenerate bilinear form

Definition

Let L/KL/K be a finite extension and let TrL/K\mathrm{Tr}_{L/K} be the field trace. The trace form of L/KL/K is the map B ⁣:L×L→K,B(x,y)=TrL/K(xy).B \colon L \times L \to K, \qquad B(x, y) = \mathrm{Tr}_{L/K}(xy).

Remarks

Since TrL/K\mathrm{Tr}_{L/K} is KK-linear and multiplication is commutative and distributive, BB is a symmetric bilinear form on the KK-vector space LL. It is nondegenerate if and only if L/KL/K is separable. Its quadratic form is x↦TrL/K(x2)x \mapsto \mathrm{Tr}_{L/K}(x^2).

Example: L=ℚ(i)L = \Q(i) over K=ℚK = \Q. Check: (a+bi)(c+di)=(ac−bd)+(ad+bc)i(a + bi)(c + di) = (ac - bd) + (ad + bc)i, so B(a+bi,c+di)=2(ac−bd)B(a + bi, c + di) = 2(ac - bd); in the basis 1,i1, i the matrix of BB is (200−2)\begin{pmatrix} 2 & 0 \\ 0 & -2 \end{pmatrix}, which is invertible.

Non-example of nondegeneracy: let K=𝔽p(t)K = \F_p(t) and L=K(s)L = K(s) with sp=ts^p = t. The trace of every element of LL is 00, so the trace form of L/KL/K is identically zero and degenerate, as expected for an inseparable extension.

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Group, concept, no evidence levelGroupconceptMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionRing, definition, no evidence levelRingdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionBilinear form, definition, no evidence levelBilinear formdefinitionLinear map, definition, no evidence levelLinear mapdefinitionCommutative 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 ringdefinitionTrace of a matrix, definition, no evidence levelTrace of a matrixdefinitionNondegenerate bilinear form, definition, no evidence levelNondegenerate bilinea…definitionSymmetric bilinear form, definition, no evidence levelSymmetric bilinear fo…definitionEndomorphism, definition, no evidence levelEndomorphismdefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionTrace of a linear map, definition, no evidence levelTrace of a linear mapdefinitionLeading 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 elementdefinitionDegree of a field extension, definition, no evidence levelDegree of a field ext…definitionMinimal polynomial of an algebraic element, definition, no evidence levelMinimal polynomial of…definitionFinite extension, definition, no evidence levelFinite extensiondefinitionField trace, definition, no evidence levelField tracedefinitionTrace form, definition, no evidence levelTrace formdefinition
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. Matrix definition
  3. Vector space definition
  4. Ring definition
  5. Square matrix definition
  6. Bilinear form definition
  7. Linear map definition
  8. Commutative ring definition
  9. Polynomial ring definition
  10. Subring definition
  11. Unit of a ring definition
  12. Trace of a matrix definition
  13. Nondegenerate bilinear form definition
  14. Symmetric bilinear form definition
  15. Endomorphism definition
  16. Degree of a polynomial definition
  17. Polynomial evaluation definition
  18. Field definition
  19. Trace of a linear map definition
  20. Leading coefficient definition
  21. Root of a polynomial definition
  22. Subfield definition
  23. Monic polynomial definition
  24. Field extension definition
  25. Algebraic element definition
  26. Degree of a field extension definition
  27. Minimal polynomial of an algebraic element definition
  28. Finite extension definition
  29. Field trace definition
  30. Trace form definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Field_trace
About this page
TagDQWX2Y8EA5
URIm2://DQWX2Y8EA5
Kinddefinition
WritingW0 stub
MSC12F10, 11E04, 11R29
DefinesTrace form
UsesFinite extension, Field trace, Symmetric bilinear form, Nondegenerate bilinear form
Changed2 November 2026

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