Hermitian dot product (finite fields)
Let be an http://planetmath.org/node/4703even http://planetmath.org/node/438prime power (in particular, is a square) and
the finite field with elements. Then
is a subfield
of . The
of an element is defined by
the
-th power Frobenius map
The has properties similar to the complex conjugate. Let
, then
-
1.
,
-
2.
,
-
3.
.
Properties 1 and 2 hold because the Frobenius map is a
http://planetmath.org/node/1011homomorphism.
Property 3 holds because of the identity
which
holds for any in any finite field with elements.
See also http://planetmath.org/node/2893finite field.
Now let be the -dimensional vector space over
, then the Hermitian dot product of two vectors
is
Again, this kind of Hermitian dot product has properties similar to
Hermitian inner products on complex vector spaces. Let
and , then
-
1.
(linearity)
-
2.
-
3.
Property 3 follows since divides (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 | conjugate |
Defines | conjugation![]() |