Diophantine avoidance, primitive elements, and normal basis theorem (Sehun Jeong, CMC)

Diophantine avoidance has been studied by several authors in recent years. This refers to effective results on existence of points of bounded size in a given algebraic set avoiding some specified subsets. The famous primitive element theorem states that every number field K is of the form Q(a) for some element a in K. From the proof of this theorem, it is clear that not only there are infinitely many such primitive elements in K, but in fact most elements in K are primitive. One natural question to ask is about finding a primitive element of small “size”, where we use a height function to measure size. We discuss the conjecture of Ruppert and known results in this direction, as well as our recent work on this problem. In addition, we provide the standard and effective version of normal basis theorem, obtaining an explicit bound in terms of the degree and discriminant of K, where K/Q is a Galois extension. At the end, we discuss a particularly good bound in the case of prime degree. This is joint work with Lenny Fukshansky.