Indian Journal of Pure and Applied Mathematics | 2021
An effective version of the primitive element theorem
Abstract
Let $$\\alpha $$\n and $$\\beta $$\n be two algebraic numbers, $$F={{\\mathbb {Q}}}(\\alpha ,\\beta )$$\n and $$d=[F:{{\\mathbb {Q}}}] \\ge 2$$\n . By the primitive element theorem, for all but finitely many rational numbers r we have $$F={{\\mathbb {Q}}}(\\alpha +r\\beta )$$\n . A straightforward argument implies that the number of exceptional r, namely, those $$r \\in {{\\mathbb {Q}}}$$\n for which $${{\\mathbb {Q}}}(\\alpha +r\\beta )$$\n is a proper subfield of F, is at most $$(d-1)^2$$\n . We show that the number of exceptional r is at most d. On the other hand, we give an example showing the number of exceptional r can be greater than $$\\big (\\frac{\\log d}{\\log \\log d}\\big )^2$$\n for infinitely many $$d \\in {\\mathbb {N}}$$\n .