Abstract

Letq>1 be a prime power,m>1 an integer,GF(q m) andGF (q) the Galois fields of orderq m andq, respectively. We show that the different module structures of (GF(q m), +) arising from the intermediate fields of the field extensionGF(q m) overGF (q), can be studied simultaneously with the help of some basic properties of cyclotomic polynomials. The results can be generalized to finite cyclic Galois extensions over arbitrary fields.In 1986, D. Blessenohl and K. Johnsen proved that there exist elements inGF(q m) which generate normal bases inGF(q m) overany intermediate fieldGF(q d) ofGF(q m) overGF(q). Such elements are called completely free inGF(q m) overGF(q). Using our ideas, we give a detailed and constructive proof of the most difficult part of that theorem, i.e., the existence of completely free elements inGF(q m), overGF(q) provided thatm is a prime power. The general existence problem of completely free elements is easily reduced to this special case.Furthermore, we develop a recursive formula for the number of completely free elements inGF(q m) overGF(q) in the case wherem is a prime power.

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