Constructive Results
Generally, the set of all primitive elements for a finite separable extension L / K is the complement of a finite collection of proper K-subspaces of L, namely the intermediate fields. This statement says nothing for the case of finite fields, for which there is a computational theory dedicated to finding a generator of the multiplicative group of the field (a cyclic group), which is a fortiori a primitive element. Where K is infinite, a pigeonhole principle proof technique considers the linear subspace generated by two elements and proves that there are only finitely many linear combinations
with c in K in it, that fail to generate the subfield containing both elements. This is almost immediate as a way of showing how Artin's result implies the classical result, and a bound for the number of exceptional c in terms of the number of intermediate fields results (this number being something that can be bounded itself by Galois theory and a priori). Therefore in this case trial-and-error is a possible practical method to find primitive elements. See the Example.
Read more about this topic: Primitive Element Theorem
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