Galois ring
The Galois ring is [a5] the unique Galois extension of
of degree
. For instance
is
and
is
. Generalizing finite fields (cf. Finite field), these rings find applications in similar areas: linear recurrences [a1], [a6], [a7], cyclic codes [a9], [a2] [a3], association schemes [a10], and character sums [a10], [a4]. For a connection with Witt rings see [a8] (cf. also Witt ring). Two different constructions of these rings are given below: bottom-up, starting from a finite field, and top-down, starting from a local field.
Bottom up.
This is the first and the most algorithmic one. Let . Pick an irreducible monic primitive polynomial
of degree
, as in the standard construction of
from
, and lift it to a polynomial
over
in such a way that the nice finite field property (
divides
) still holds. In the language of linear recurrences (or linear feedback shift registers), one has lifted an
-sequence of period
over
into a linear recurrence over
of the same period. This is construction
of [a1]. Note that an arbitrary lift will lead to multiplying the period by a power of
, as in construction
of [a1]. For example,
,
gives a period
and not
. Now, let
![]() |
Top down.
This -adic approach was introduced in print in [a4] but was implicitly known to M. Yamada [a10], who used the term "Teichmüller" , as in
-adic analysis, and also to E. Spiegel [a9]. Denote by
the ring of
-adic integers, best viewed as the set of formal expansions in powers of
with coefficients in the residue field
. Then
. For higher values of
one considers the unramified extension of
generated by
(an
-th root of unity) and its ring of integers
. Let
denote the set of
roots of
over this latter ring. This set of so-called Teichmüller representatives reduces modulo
to
. The ring of integers of the
-adic field admits the following expansion:
, which converges in the sense of the
-adic valuation. Modulo
this yields
![]() |
Multiplicative structure.
The ring comprises units
and zero divisors
. The multiplicative group
is the direct product of
by the group of so-called principal units
. The group of principal units is isomorphic, for
or
, to the additive group of
. The Galois group of
over
is isomorphic to the Galois group of
over
and therefore cyclic of order
.
References
[a1] | S. Boztas, A.R. Hammons, P.V. Kumar, "![]() |
[a2] | A. Bonnecaze, P. Solé, A.R. Calderbank, "Quaternary construction of unimodular lattices" IEEE Inform. Th. , 41 (1995) pp. 366–376 |
[a3] | A.R. Hammons, P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, P. Solé, "The ![]() |
[a4] | V. Kumar, T. Helleseth, R.A. Calderbank, "An upper bound for Weil-type exponential sums over Galois rings and applications" IEEE Inform. Th. , 41 (1995) |
[a5] | B.R. MacDonald, "Finite rings with identity" , M. Dekker (1974) |
[a6] | P. Solé, "A quaternary cyclic code and a family of quaternary sequences with low correlation" G. Cohen (ed.) J. Wolfmann (ed.) , Coding Theory and Applications , Lecture Notes in Computer Science , 388 , Springer (1989) pp. 193–201 |
[a7] | P. Udaya, M.U. Siddiqui, "Optimal biphase sequences with large linear complexity derived from sequences over ![]() |
[a8] | A.G. Shanbag, P.V. Kumar, T. Helleseth, "An upperbound for the extended Kloosterman sums over Galois rings" , Finite Fields and Applications (to appear) |
[a9] | E. Spiegel, "Codes over ![]() |
[a10] | M. Yamada, "Distance regular graphs of girth ![]() ![]() |
Galois ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Galois_ring&oldid=47033