View source for Galois ring

The Galois ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g1100401.png" /> is [[#References|[a5]]] the unique [[Galois extension|Galois extension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g1100402.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g1100403.png" />. For instance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g1100404.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g1100405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g1100406.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g1100407.png" />. Generalizing finite fields (cf. [[Finite field|Finite field]]), these rings find applications in similar areas: linear recurrences [[#References|[a1]]], [[#References|[a6]]], [[#References|[a7]]], cyclic codes [[#References|[a9]]], [[#References|[a2]]] [[#References|[a3]]], association schemes [[#References|[a10]]], and character sums [[#References|[a10]]], [[#References|[a4]]]. For a connection with Witt rings see [[#References|[a8]]] (cf. also [[Witt ring|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|local field]].

==Bottom up.==
This is the first and the most algorithmic one. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g1100408.png" />. Pick an irreducible monic primitive [[Polynomial|polynomial]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g1100409.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004010.png" />, as in the standard construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004011.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004012.png" />, and lift it to a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004013.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004014.png" /> in such a way that the nice finite field property (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004015.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004016.png" />) still holds. In the language of linear recurrences (or linear feedback shift registers), one has lifted an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004017.png" />-sequence of period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004018.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004019.png" /> into a linear recurrence over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004020.png" /> of the same period. This is construction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004022.png" /> of [[#References|[a1]]]. Note that an arbitrary lift will lead to multiplying the period by a power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004023.png" />, as in construction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004025.png" /> of [[#References|[a1]]]. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004027.png" /> gives a period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004028.png" /> and not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004029.png" />. Now, let

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==Top down.==
This <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004031.png" />-adic approach was introduced in print in [[#References|[a4]]] but was implicitly known to M. Yamada [[#References|[a10]]], who used the term  "Teichmüller" , as in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004032.png" />-adic analysis, and also to E. Spiegel [[#References|[a9]]]. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004033.png" /> the ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004034.png" />-adic integers, best viewed as the set of formal expansions in powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004035.png" /> with coefficients in the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004036.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004037.png" />. For higher values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004038.png" /> one considers the unramified extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004039.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004040.png" /> (an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004041.png" />-th root of unity) and its ring of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004042.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004043.png" /> denote the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004044.png" /> roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004045.png" /> over this latter ring. This set of so-called Teichmüller representatives reduces modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004046.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004047.png" />. The ring of integers of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004048.png" />-adic field admits the following expansion: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004049.png" />, which converges in the sense of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004050.png" />-adic valuation. Modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004051.png" /> this yields

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==Multiplicative structure.==
The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004053.png" /> comprises units <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004054.png" /> and zero divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004055.png" />. The multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004056.png" /> is the direct product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004057.png" /> by the group of so-called principal units <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004058.png" />. The group of principal units is isomorphic, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004059.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004060.png" />, to the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004061.png" />. The Galois group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004062.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004063.png" /> is isomorphic to the Galois group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004064.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004065.png" /> and therefore cyclic of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004066.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Boztas,   A.R. Hammons,   P.V. Kumar,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004067.png" />-phase sequence with near optimum correlation properties"  ''IEEE Inform. Th.'' , '''38'''  (1992)  pp. 1101–1113</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Bonnecaze,   P. Solé,   A.R. Calderbank,   "Quaternary construction of unimodular lattices"  ''IEEE Inform. Th.'' , '''41'''  (1995)  pp. 366–376</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.R. Hammons,   P.V. Kumar,   A.R. Calderbank,   N.J.A. Sloane,   P. Solé,   "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004068.png" />-linearity of Kerdock, Preparata, Goethals, and related codes"  ''IEEE Trans. Information Th.'' , '''40'''  (1994)  pp. 301–319</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.R. MacDonald,   "Finite rings with identity" , M. Dekker  (1974)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P. Udaya,   M.U. Siddiqui,   "Optimal biphase sequences with large linear complexity derived from sequences over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004069.png" />"  ''IEEE Inform. Th.'' , '''IT–42'''  (1996)  pp. 202–216</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A.G. Shanbag,   P.V. Kumar,   T. Helleseth,   "An upperbound for the extended Kloosterman sums over Galois rings" , ''Finite Fields and Applications''  (to appear)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E. Spiegel,   "Codes over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004070.png" /> revisited"  ''Inform. and Control'' , '''37'''  (1978)  pp. 100–104</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  M. Yamada,   "Distance regular graphs of girth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004071.png" /> over an extension ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110040/g11004072.png" />"  ''Graphs and Combinatorics'' , '''6'''  (1980)  pp. 381–384</TD></TR></table>