Hermitian dot product (finite fields)

Let q be an http://planetmath.org/node/4703even http://planetmath.org/node/438prime power (in particular, q is a square) and 𝔽q the finite fieldMathworldPlanetmath with q elements. Then 𝔽q is a subfieldMathworldPlanetmath of 𝔽q. The k¯ of an element k𝔽q is defined by the q-th power Frobenius mapPlanetmathPlanetmath


The has properties similar to the complex conjugateMathworldPlanetmath. Let k1,k2𝔽q, then

  1. 1.


  2. 2.


  3. 3.


Properties 1 and 2 hold because the Frobenius map is a http://planetmath.org/node/1011homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. Property 3 holds because of the identityPlanetmathPlanetmath kq=k which holds for any k in any finite field with q elements. See also http://planetmath.org/node/2893finite field.

Now let 𝔽qn be the n-dimensional vector spaceMathworldPlanetmath over 𝔽q, then the Hermitian dot product of two vectors (u1,,un),(v1,,vn)𝔽qn is


Again, this kind of Hermitian dot product has properties similar to Hermitian inner productsMathworldPlanetmath on complex vector spaces. Let k1,k2𝔽q and v1,v2,v,,w𝔽qn, then

  1. 1.

    (k1v1+k2v2)w=k1(v1w)+k2(v2w) (linearity)

  2. 2.


  3. 3.


Property 3 follows since q+1 divides q-1 (see http://planetmath.org/node/2893finite field).

Title Hermitian dot product (finite fields)
Canonical name HermitianDotProductfiniteFields
Date of creation 2013-03-22 15:13:26
Last modified on 2013-03-22 15:13:26
Owner GrafZahl (9234)
Last modified by GrafZahl (9234)
Numerical id 7
Author GrafZahl (9234)
Entry type Definition
Classification msc 11E39
Classification msc 12E20
Synonym Hermitian dot product
Related topic FiniteField
Defines conjugatePlanetmathPlanetmathPlanetmath (finite fields)
Defines conjugationMathworldPlanetmath (finite fields)