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Dual basis in a field extension — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Dual_basis_in_a_field_extension
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-15T16:14:39.486Z
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Dual basis in a field extension — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Dual_basis_in_a_field_extension (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_d4dedcf189da (Metraipolis archive dc_d4dedcf189da, sha256 ab6d3d379d5d02847f7eb99366b6a91d18b3e941956bca004cef0686e00c6046)

BibTeX

@misc{dcd4dedcf189da,
  title = {Dual basis in a field extension — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Dual_basis_in_a_field_extension},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_d4dedcf189da; sha256 ab6d3d379d5d02847f7eb99366b6a91d18b3e941956bca004cef0686e00c6046},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Dual_basis_in_a_field_extension. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite field extension L/K, by using the field trace. This requires the property that the field trace TrL/K provides a non-degenerate quadratic form over K. This is true if L is separable over K; it is always true if K is a perfect field, including when K is finite or of characteristic zero. A dual basis is not a specific basis like the polynomial basis or the normal basis; rather for any given basis, it is another associated basis which is useful for computations. We say that two bases of a finite field L = GF ( p n ) {\displaystyle L={\text{GF}}(p^{n})} over K = GF ( p ) = Z / p Z {\displaystyle K={\text{GF}}(p)=\mathbb {Z} /p\mathbb {Z} } , B 1 = { α 0 , α 1 , … , α n − 1 } , B 2 = { β 0 , β 1 , … , β n − 1 } , {\displaystyle B_{1}=\{\alpha _{0},\alpha _{1},\ldots ,\alpha _{n-1}\},\qquad B_{2}=\{\beta _{0},\beta _{1},\ldots ,\beta _{n-1}\},} are dual to each other provided Tr ( α i ⋅ β j ) = { 1 if i = j 0 if i ≠ j . {\displaystyle \operatorname {Tr} (\alpha _{i}\cdot \beta _{j})={\begin{cases}1&{\text{if }}\ i=j\\0&{\text{if }}\ i\neq j.\end{cases}}} Here the trace of a value in GF(pm) can be calculated as follows: Tr ( γ ) = ∑ i = 0 n − 1 γ p i {\displaystyle \operatorname {Tr} (\gamma )=\sum _{i=0}^{n-1}\gamma ^{p^{i}}} Using a dual basis can provide a way to easily communicate between devices that use different bases, rather than having to explicitly convert between bases using the change of bases formula. Furthermore, if a dual basis is implemented then conversion from an element in the original basis to the dual basis can be accomplished with multiplication by the multiplicative identity (usually 1).

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; categoria «Linear algebra»).
Fonte: Wikipedia in inglese, voce «Dual basis in a field extension» — https://en.wikipedia.org/wiki/Dual_basis_in_a_field_extension
Wikidata Q5310185 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q5310185
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Dual_basis_in_a_field_extension?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1362383856, consultata il 2026-10-15T15:42:02.506Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: a09b419c99a38030940b3affc8ead1b341da65e0b58c880cf14aa4c0b2656bb5