PlanetMath: Hermitian dot product (finite fields)

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Hermitian dot product (finite fields)

(Definition)

Let be an even prime power (in particular, is a square) and $\mathbb{F}_q$ the finite field with elements. Then $\mathbb{F}_{\sqrt{q}}$ is a subfield of $\mathbb{F}_q$ . The conjugate $\overline {k}$ of an element $k\in\mathbb{F}_q$ is defined by the $\sqrt{q}$ -th power Frobenius map

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

The conjugate has properties similar to the complex conjugate. Let $k_1,k_2\in\mathbb{F}_q$ , then

$\overline {k_1+k_2}=\overline {k_1}+\overline {k_2}$ ,
$\overline {k_1k_2}=\overline {k_1}\,\overline {k_2}$ ,
$\overline {\overline {k_1}}=k_1$ .

Properties 1 and 2 hold because the Frobenius map is a homomorphism. Property 3 holds because of the identity

which holds for any

in any finite field with

elements. See also finite field.

Now let $\mathbb{F}_q^n$ be the -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

$\displaystyle (u_1,\ldots,u_n)\cdot(v_1,\ldots,v_n):=\sum\limits _{i=1}^nu_i\overline {v_i}.$

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

$(k_1v_1+k_2v_2)\cdot w=k_1(v_1\cdot w)+k_2(v_2\cdot w)$ (linearity)
$v\cdot w=\overline {w\cdot v}$
$v\cdot v\in\mathbb{F}_{\sqrt{q}}$

Property 3 follows since $\sqrt{q}+1$ divides

(see finite field).

"Hermitian dot product (finite fields)" is owned by GrafZahl.

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See Also: finite field

Other names:

Hermitian dot product

Also defines:

conjugate (finite fields), conjugation (finite fields)

Keywords:

inner product

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Cross-references: divides, complex, Hermitian inner products, vectors, vector space, identity, complex conjugate, similar, properties, Frobenius map, subfield, finite field, square
There is 1 reference to this entry.

This is version 4 of Hermitian dot product (finite fields), born on 2005-05-01, modified 2005-05-03.
Object id is 6988, canonical name is HermiteanDotProductFiniteFields.
Accessed 3334 times total.

Classification:

AMS MSC:	11E39 (Number theory :: Forms and linear algebraic groups :: Bilinear and Hermitian forms)
	12E20 (Field theory and polynomials :: General field theory :: Finite fields )

Pending Errata and Addenda

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