Extensions of normal bases and completely basic fields
Carl C. Faith · Transactions of the American Mathematical Society · 1957
A finite, separable, normal extension %/a, with Galois group (M = (S1, S2, * * *, S.) always possesses a basis of the form wse, wS2, . . ., wSn, wEz9, called a normal basis of 9/W. Then, w generates a normal basis of T/, or w is a normal basis element of 91/a. Many proofs of the existence of such a basis are available('). Now let A be any intermediate field, 91DADa. We raise the following question: Does there exist a normal basis of 9/A which is extendable to a normal basis of %/s, or equivalently, is it possible for an element w to generate a normal basis in both extensions? It is shown in Chapter I that the answer to this question is in the affirmative. Moreover, we show (Theorem 1.1) that one may choose w (when j is infinite) independently of the intermediate field A, i.e., so that w generates a normal basis of 9/F, for every intermediate field F. We call elements having this latter property completely basic elements of 9/j. Every completely basic element of %/a is a normal basis element of this extension. It is natural to inquire into the special character of normal basis elements by asking whether each normal basis element is completely basic. Although in general the answer to this question is negative (certain cyclotomic extensions provide counterexamples), we are able to establish the existence of a significant class E of those normal extensions, called completely basic extensions, for which every normal basis element is completely basic. For these extensions every normal basis element of )/j is the extension of a normal basis of 9/A, for any intermediate field A. However, not every normal basis of W/A is extendable to normal basis of 9/~ when A(Theorem 1.7). In Chapter I, a new characterization of the normal basis elements in cyclic extensions is used, with practically no additional machinery, to establish the existence of various completely basic extensions, including the cyclic Kummer extensions. The extension of this latter result to an arbitrary Kummer extension f/ is the main ingredient of Chapter II. The proof that f/(zS does not use the fact that every cyclic Kummer extension (3/j belongs to E,, even though it is known that f is the direct product over a of