Universal Generators for Primary Closures of Galois Fields

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.