We present a short proof, based on Cogalois Theory, of a result due to Acosta de Orozco and V´elez characterizing separable simple radical field extensions with the unique subfield property, and prove that these extensions are precisely the simple G–Cogalois extensions with a cyclic Kneser group. | Turk J Math 26 (2002) , 433 – 445. ¨ ITAK ˙ c TUB Field Extensions Having the Unique Subfield Property, and G-Cogalois Extensions Toma Albu∗ Abstract We present a short proof, based on Cogalois Theory, of a result due to Acosta de Orozco and V´elez (1982, J. Number Theory 15, 388-405) characterizing separable simple radical field extensions with the unique subfield property, and prove that these extensions are precisely the simple G–Cogalois extensions with a cyclic Kneser group. Key words and phrases: Field extension, separable extension, simple extension, radical extension, G–Cogalois extension, unique subfield property, classical Kummer extension. Introduction The aim of this paper is to investigate via Cogalois Theory field extensions with the unique subfield property considered by V´elez [10], [11] and by Acosta de Orozco and V´elez [1]. We present in this framework an alternative proof of the Acosta de Orozco–V´elez Criterion [1] characterizing separable simple radical extensions with the unique subfield property. We show that a separable simple radical extension has the unique subfield property if and only if it is G–Cogalois with cyclic Kneser group. Using this fact, we retrieve immediately a result of V´elez [10]. 2000 Mathematics Subject Classification: 12E30, 12F05, 12F10, 12F99. work was carried out during a stay of the author at the Atilim University, Ankara in the academic year 2001-2002. He would like to thank the University for hospitality and financial support. ∗ This 433 ALBU 0. Preliminaries Throughout this paper F denotes a fixed field with characteristic exponent e(F ) and Ω a fixed algebraic closure of F . Any algebraic extension of F is supposed to be a subfield of Ω. For an arbitrary nonempty subset S of Ω and a natural number n ≥ 1 we shall use the following notation: S ∗ := S \ {0}, S n := { xn | x ∈ S }, µn (S) := { x ∈ S | xn = 1 }. We denote by N the set {0, 1, 2, . . .} of all natural numbers, by N∗ the set N {0} of all .
