Monogenic Cyclic Cubic Trinomials
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909431700127744 |
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| author | Jones, Lenny |
| author_facet | Jones, Lenny |
| contents | A series of recent articles has shown that there exist only three monogenic cyclic quartic trinomials in ${\mathbb Z}[x]$, and they are all of the form $x^4+bx^2+d$. In this article, we conduct an analogous investigation for cubic trinomials in ${\mathbb Z}[x]$. Two irreducible cyclic cubic trinomials are said to be equivalent if their splitting fields are equal. We show that there exist two infinite families of non-equivalent monogenic cyclic cubic trinomials of the form $x^3+Ax+B$. We also show that there exist exactly four monogenic cyclic cubic trinomials of the form $x^3+Ax^2+B$, all of which are equivalent to $x^3-3x+1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13075 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monogenic Cyclic Cubic Trinomials Jones, Lenny Number Theory A series of recent articles has shown that there exist only three monogenic cyclic quartic trinomials in ${\mathbb Z}[x]$, and they are all of the form $x^4+bx^2+d$. In this article, we conduct an analogous investigation for cubic trinomials in ${\mathbb Z}[x]$. Two irreducible cyclic cubic trinomials are said to be equivalent if their splitting fields are equal. We show that there exist two infinite families of non-equivalent monogenic cyclic cubic trinomials of the form $x^3+Ax+B$. We also show that there exist exactly four monogenic cyclic cubic trinomials of the form $x^3+Ax^2+B$, all of which are equivalent to $x^3-3x+1$. |
| title | Monogenic Cyclic Cubic Trinomials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2412.13075 |