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.