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 $\mathbb{F}_{q}$ the finite field with $q$ elements. Then $\mathbb{F}_{\sqrt{q}}$ is a subfield of $\mathbb{F}_{q}$. The $\overline{k}$ of an element $k\in\mathbb{F}_{q}$ is defined by the $\sqrt{q}$-th power Frobenius map

 $\overline{k}:=\operatorname{Frob}_{\sqrt{q}}(k)=k^{\sqrt{q}}.$

The has properties similar to the complex conjugate. Let $k_{1},k_{2}\in\mathbb{F}_{q}$, then

1. 1.

$\overline{k_{1}+k_{2}}=\overline{k_{1}}+\overline{k_{2}}$,

2. 2.

$\overline{k_{1}k_{2}}=\overline{k_{1}}\,\overline{k_{2}}$,

3. 3.

$\overline{\overline{k_{1}}}=k_{1}$.

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

Now let $\mathbb{F}_{q}^{n}$ be the $n$-dimensional vector space over $\mathbb{F}_{q}$, then the Hermitian dot product of two vectors $(u_{1},\ldots,u_{n}),(v_{1},\ldots,v_{n})\in\mathbb{F}_{q}^{n}$ is

 $(u_{1},\ldots,u_{n})\cdot(v_{1},\ldots,v_{n}):=\sum\limits_{i=1}^{n}u_{i}% \overline{v_{i}}.$

Again, this kind of Hermitian dot product has properties similar to Hermitian inner products on complex vector spaces. Let $k_{1},k_{2}\in\mathbb{F}_{q}$ and $v_{1},v_{2},v,,w\in\mathbb{F}_{q}^{n}$, then

1. 1.

$(k_{1}v_{1}+k_{2}v_{2})\cdot w=k_{1}(v_{1}\cdot w)+k_{2}(v_{2}\cdot w)$ (linearity)

2. 2.

$v\cdot w=\overline{w\cdot v}$

3. 3.

$v\cdot v\in\mathbb{F}_{\sqrt{q}}$

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

Title Hermitian dot product (finite fields) HermitianDotProductfiniteFields 2013-03-22 15:13:26 2013-03-22 15:13:26 GrafZahl (9234) GrafZahl (9234) 7 GrafZahl (9234) Definition msc 11E39 msc 12E20 Hermitian dot product FiniteField conjugate (finite fields) conjugation (finite fields)