Eisenstein-prime Obstruction Sieve for Monogenicity
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908827531608064 |
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| author | Nguyen-Dang, Khai-Hoan |
| author_facet | Nguyen-Dang, Khai-Hoan |
| contents | Alpöge--Bhargava--Shnidman showed that even a strengthened \emph{no local obstruction} condition for monogenicity does not force a global power integral basis: in the full spaces of cubic and quartic fields, a positive proportion are non-monogenic yet satisfy this ABS fixed-sign condition. This raises a natural family-level question: does the same phenomenon persist inside one-parameter families, where the local structure varies in a highly constrained way?
In this paper we answer this in the negative for the pure fields $K_m=\mathbb Q(α)$ with $α^n=m$ ($n\ge 4$) and $m$ square-free. Writing $g(m)=[\mathcal O_{K_m}:\mathbb Z[α]]$, we prove that the set of square-free $m$ for which $g(m)>1$ but $K_m$ has no ABS local obstruction has natural density $0$. Consequently, in the pure family monogenicity and $α$--monogenicity have the same natural density. The proof isolates a reusable mechanism, which we call the Eisenstein-prime obstruction sieve. The argument is packaged in an abstract template and transfers to other Eisenstein parameter families. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10488 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Eisenstein-prime Obstruction Sieve for Monogenicity Nguyen-Dang, Khai-Hoan Number Theory Alpöge--Bhargava--Shnidman showed that even a strengthened \emph{no local obstruction} condition for monogenicity does not force a global power integral basis: in the full spaces of cubic and quartic fields, a positive proportion are non-monogenic yet satisfy this ABS fixed-sign condition. This raises a natural family-level question: does the same phenomenon persist inside one-parameter families, where the local structure varies in a highly constrained way? In this paper we answer this in the negative for the pure fields $K_m=\mathbb Q(α)$ with $α^n=m$ ($n\ge 4$) and $m$ square-free. Writing $g(m)=[\mathcal O_{K_m}:\mathbb Z[α]]$, we prove that the set of square-free $m$ for which $g(m)>1$ but $K_m$ has no ABS local obstruction has natural density $0$. Consequently, in the pure family monogenicity and $α$--monogenicity have the same natural density. The proof isolates a reusable mechanism, which we call the Eisenstein-prime obstruction sieve. The argument is packaged in an abstract template and transfers to other Eisenstein parameter families. |
| title | Eisenstein-prime Obstruction Sieve for Monogenicity |
| topic | Number Theory |
| url | https://arxiv.org/abs/2602.10488 |