Counting number fields of fixed degree by their smallest defining polynomial
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915780590829568 |
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| author | Arango-Piñeros, Santiago Gundlach, Fabian Oliver, Robert J. Lemke McGown, Kevin J. Sawin, Will López, Allechar Serrano Shankar, Arul Varma, Ila |
| author_facet | Arango-Piñeros, Santiago Gundlach, Fabian Oliver, Robert J. Lemke McGown, Kevin J. Sawin, Will López, Allechar Serrano Shankar, Arul Varma, Ila |
| contents | When do two irreducible polynomials with integer coefficients
define the same number field? One can define an action of
$\mathrm{GL}_2 \times \mathrm{GL}_1$ on the space of polynomials of degree $n$ so that for any two
polynomials $f$ and $g$ in the same orbit, the roots of $f$ may be expressed
as rational linear transformations of the roots of $g$; thus, they generate
the same field. In this article, we show that almost all polynomials of
degree $n$ with size at most $X$ can only define the same number field as
another polynomial of degree $n$ with size at most $X$ if they lie in the
same orbit for this group action. (Here we measure the size of polynomials by
the greatest absolute value of their coefficients.)
This improves on work of Bhargava, Shankar, and Wang, who proved a similar
statement for a positive proportion of polynomials. Using this result, we
prove that the number of degree $n$ fields such that the smallest polynomial
defining the field has size at most $X$ is asymptotic to a constant times
$X^{n+1}$ as long as $n\geq 3$. For $n = 2$, we obtain a precise asymptotic of
the form $\frac{27}{π^2} X^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06943 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting number fields of fixed degree by their smallest defining polynomial Arango-Piñeros, Santiago Gundlach, Fabian Oliver, Robert J. Lemke McGown, Kevin J. Sawin, Will López, Allechar Serrano Shankar, Arul Varma, Ila Number Theory When do two irreducible polynomials with integer coefficients define the same number field? One can define an action of $\mathrm{GL}_2 \times \mathrm{GL}_1$ on the space of polynomials of degree $n$ so that for any two polynomials $f$ and $g$ in the same orbit, the roots of $f$ may be expressed as rational linear transformations of the roots of $g$; thus, they generate the same field. In this article, we show that almost all polynomials of degree $n$ with size at most $X$ can only define the same number field as another polynomial of degree $n$ with size at most $X$ if they lie in the same orbit for this group action. (Here we measure the size of polynomials by the greatest absolute value of their coefficients.) This improves on work of Bhargava, Shankar, and Wang, who proved a similar statement for a positive proportion of polynomials. Using this result, we prove that the number of degree $n$ fields such that the smallest polynomial defining the field has size at most $X$ is asymptotic to a constant times $X^{n+1}$ as long as $n\geq 3$. For $n = 2$, we obtain a precise asymptotic of the form $\frac{27}{π^2} X^2$. |
| title | Counting number fields of fixed degree by their smallest defining polynomial |
| topic | Number Theory |
| url | https://arxiv.org/abs/2602.06943 |