Universal Generators for Primary Closures of Galois Fields

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.