Universal Generators for Primary Closures of Galois Fields

Dirk Hachenberger ()
Additional contact information
Dirk Hachenberger: Institut für Mathematik der Universität Augsburg

A chapter in Finite Fields and Applications, 2001, pp 208-223 from Springer

Abstract: Abstract If $$ \bar{F} $$ is an algebraic closure of a Galois field F, then for each integer n ≥ 1 there is exactly one subfield E n of $$ \bar{F} $$ containing F and having degree n over F. For a prime number r, we consider the r-primary closure $$ {\bar{F}_{r}}: = \bigcup {_{{m \geqslant 0}}{E_{{{r^{m}}}}}} $$ over F and prove, under the assumption that r ≥ 7, but without any restriction on the cardinality q of F, the existence of a universal generator for $$ {\bar{F}_{r}} $$ over F: this is a sequence $$ w = {({w_{{{r^{m}}}}})_{{m \geqslant 0}}} $$ in $$ {\bar{F}_{r}} $$ which satisfies all the following properties: (1) $$ {w_{{{r^{m}}}}} $$ is a. primitive element of $$ {E_{{{r^{m}}}}} $$ (for all m ≥ 0), (2) $$ {w_{{{r^{m}}}}} $$ generates a normal basis for $$ {E_{{{r^{m}}}}} $$ over F (for all m ≥ 0), (3) w is norm-compatible, (4) w is trace compatible. We prove furthermore that (2) can be strengthened to (2 c ) $$ {w_{{{r^{m}}}}} $$ is completely free in $$ {E_{{{r^{m}}}}} $$ over F (for all m ≥ 0),which means that $$ {w_{{{r^{m}}}}} $$ simultaneously generates a normal basis for $$ {E_{{{r^{m}}}}} $$ over $$ {E_{{{r^{i}}}}} $$ for all i = 0, 1, …, m, whence w is called a complete universal generator for $$ {\bar{F}_{r}} $$ over F. The results establish a (complete) primitive normal basis theorem for $$ {\bar{F}_{r}} $$ over F.

Keywords: Finite Field; Galois Group; Primary Closure; Prime Power; Prime Divisor (search for similar items in EconPapers)
Date: 2001
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-56755-1_17

Ordering information: This item can be ordered from
http://www.springer.com/9783642567551

DOI: 10.1007/978-3-642-56755-1_17

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().