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**1 - 4**of**4**### ON NUMBER FIELDS WITH NONTRIVIAL SUBFIELDS

"... Abstract. 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 i ..."

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Abstract. 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 = en 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. 1. Introduction and

### PROGRESS TOWARDS COUNTING D5 QUINTIC FIELDS

"... Abstract. Let N(5, D5, X) be the number of quintic number fields whose Galois closure has Galois group D5 and whose discriminant is bounded by X. By a conjecture of Malle, we expect that N(5, D5, X) ∼ C · X 12 for some constant C. The best known upper bound is N(5, D5, X) X 34+ε, and we show this ..."

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Abstract. Let N(5, D5, X) be the number of quintic number fields whose Galois closure has Galois group D5 and whose discriminant is bounded by X. By a conjecture of Malle, we expect that N(5, D5, X) ∼ C · X 12 for some constant C. The best known upper bound is N(5, D5, X) X 34+ε, and we show this could be improved by counting points on a certain variety defined by a norm equation; computer calculations give strong evidence that this number is X 23. Finally, we show how such norm equations can be helpful by reinterpreting an earlier proof of Wong on upper bounds for A4 quartic fields in terms of a similar norm equation. 1.

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"... Abstract. We show that the number of rational points of height ≤ H on a non-rational plane curve of degree d is Od(H 2/d−δ), for some δ> 0 depending only on d. The implicit constant depends only on d. This improves a result of Heath-Brown, who proved the bound O(H2/d+). We also show that one can ..."

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Abstract. We show that the number of rational points of height ≤ H on a non-rational plane curve of degree d is Od(H 2/d−δ), for some δ> 0 depending only on d. The implicit constant depends only on d. This improves a result of Heath-Brown, who proved the bound O(H2/d+). We also show that one can take δ = 1/450 in the case d = 3.