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Dual basis in a field extension

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In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite 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 can be guaranteed if the extension is separable; it is automatically true if K is a perfect field, and hence in the cases where K is finite, or of characteristic zero.

A dual basis () is not a concrete basis like the polynomial basis or the normal basis; rather it provides a way of using a second basis for computations.

Consider two bases for elements in a finite field, GF(p):

B 1 = α 0 , α 1 , … , α m − 1 {\displaystyle B_{1}={\alpha _{0},\alpha _{1},\ldots ,\alpha _{m-1}}}

and

B 2 = γ 0 , γ 1 , … , γ m − 1 {\displaystyle B_{2}={\gamma _{0},\gamma _{1},\ldots ,\gamma _{m-1}}}

then B2 can be considered a dual basis of B1 provided

Tr ⁡ ( α i ⋅ γ j ) = { 0 , if ⁡   i ≠ j 1 , otherwise {\displaystyle \operatorname {Tr} (\alpha _{i}\cdot \gamma _{j})=\left\{{\begin{matrix}0,&\operatorname {if} \ i\neq j\\1,&\operatorname {otherwise} \end{matrix}}\right.}

Here the trace of a value in GF(p) can be calculated as follows:

Tr ⁡ ( β ) = ∑ i = 0 m − 1 β p i {\displaystyle \operatorname {Tr} (\beta )=\sum _{i=0}^{m-1}\beta ^{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).

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