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2010•Unpublished venueRequires access

ON NUMBER FIELDS WITH NONTRIVIAL SUBFIELDS

Martin Widmer, Algorithmische Diophantische Probleme, Martin Widmer

Open publisher page 7 citations

Abstract

Abstract. What is the probability for a number field of composite degree en to have a nontrivial subfield? As the reader might expect the answer heavily depends on the interpretation of probability. We show that if the fields are enumerated by the smallest height of their generators the probability is zero, at least if en> 6. This is in contrast to what one expects when the fields are enumerated by the discriminant. The main result of this article is an estimate for the number of algebraic numbers of degree en and bounded height which generate a field that contains an unspecified subfield of degree e. If n> max{e2 + e, 10} we get the correct asymptotics as the height tends to infinity. 1. Introduction and

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Abstract. What is the probability for a number field of composite degree en to have a nontrivial subfield? As the reader might expect the answer heavily depends on the interpretation of probability. We show that if the fields are enumerated by the smallest height of their generators the probability is zero, at least if en> 6. This is in contrast to what one expects when the fields are enumerated by the discriminant. The main result of this article is an estimate for the number of algebraic numbers of degree en and bounded height which generate a field that contains an unspecified subfield of degree e. If n> max{e2 + e, 10} we get the correct asymptotics as the height tends to infinity. 1. Introduction and

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Available abstract

Abstract. What is the probability for a number field of composite degree en to have a nontrivial subfield? As the reader might expect the answer heavily depends on the interpretation of probability. We show that if the fields are enumerated by the smallest height of their generators the probability is zero, at least if en> 6. This is in contrast to what one expects when the fields are enumerated by the discriminant. The main result of this article is an estimate for the number of algebraic numbers of degree en and bounded height which generate a field that contains an unspecified subfield of degree e. If n> max{e2 + e, 10} we get the correct asymptotics as the height tends to infinity. 1. Introduction and

Key concepts: Mathematics, Algebraic number field, Degree (music), Infinity, Discriminant, Bounded function, Algebraic number, Zero (linguistics)

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