An effective version of the primitive element theorem

Artūras Dubickas ()
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Artūras Dubickas: Vilnius University

Indian Journal of Pure and Applied Mathematics, 2022, vol. 53, issue 3, 720-726

Abstract: Abstract Let $$\alpha $$ α and $$\beta $$ β be two algebraic numbers, $$F={{\mathbb {Q}}}(\alpha ,\beta )$$ F = Q ( α , β ) and $$d=[F:{{\mathbb {Q}}}] \ge 2$$ d = [ F : Q ] ≥ 2 . By the primitive element theorem, for all but finitely many rational numbers r we have $$F={{\mathbb {Q}}}(\alpha +r\beta )$$ F = Q ( α + r β ) . A straightforward argument implies that the number of exceptional r, namely, those $$r \in {{\mathbb {Q}}}$$ r ∈ Q for which $${{\mathbb {Q}}}(\alpha +r\beta )$$ Q ( α + r β ) is a proper subfield of F, is at most $$(d-1)^2$$ ( d - 1 ) 2 . 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 $$\big (\frac{\log d}{\log \log d}\big )^2$$ ( log d log log d ) 2 for infinitely many $$d \in {\mathbb {N}}$$ d ∈ N .

Keywords: Primitive element; Field; Separable extension; Algebraic number; 11R04; 11R21; 12F10 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s13226-021-00166-w

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