Computing primitive elements of extension fields

Abstract

Several mathematical results and new computational methods are presented for primitive elements and their minimal polynomials of algebraic extension fields. For a field Q( α 1 ,…, α t ) obtained by adjoining algebraic numbers α 1 ,… α t to the rational number field Q, it is shown that there exists at least one vector =( s 1 ,…, s t ) of integers in a specially selected set of (−1) N vectors such that s 1 α 1 + s 2 α 2 +…+ s t α t is a primitive element, where N is the degree of Q( α 1 ,…, α t ) over Q. Furthermore, a method is presented for directly calculating such a vector, that gives a primitive element. Finally, for a given polynomial f over Q, a new method is presented for computing a primitive element of the splitting field of f and its minimal polynomial over Q.