On number fields with nontrivial subfields

Details On number fields with nontrivial subfields On number fields with nontrivial subfields

2012-04-16

arxiv.org/abs/1204.3575v1

Int. J. Number Theory 7, (2011), no. 3, 695-720

Mathematics

Number Theory

Scientific paper

10.1142/S1793042111004204

What is the probability for a number field of composite degree $d$ 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 $d>6$. 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 $d=e n$ and bounded height which generate a field that contains an unspecified subfield of degree $e$. If $n>\max\{e^2+e,10\}$ we get the correct asymptotics as the height tends to infinity.

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