Let p be an odd prime number, and F a number field. We show that when F/Q is unramified at p, any tame cyclic extension N/F of degree p has a normal integral basis if the pushed up extension $N(\zeta_p)/F(\zeta_p)$ has a normal integral basis.
"Note on Galois descent of a normal integral basis of acyclic extension of degree p." Proc. Japan Acad. Ser. A Math. Sci. 85 (10) 160 - 162, December 2009. https://doi.org/10.3792/pjaa.85.160