| Title:
|
Monogenic even sextic trinomials and their Galois groups (English) |
| Author:
|
Jones, Lenny |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2026 |
| Pages:
|
19-41 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $f(x)=x^6+Ax^{2k}+B\in {\mathbb{Z}}[x]$, with $A\ne 0$ and $k\in \lbrace 1,2\rbrace $. We say that $f(x)$ is monogenic if $f(x)$ is irreducible over ${\mathbb{Q}}$ and $\lbrace 1,\theta ,\theta ^2,\theta ^3,\theta ^4,\theta ^{5}\rbrace $ is a basis for the ring of integers of ${\mathbb{Q}}(\theta )$, where $f(\theta )=0$. For each value of $k$ and each possible Galois group $G$ of $f(x)$ over ${\mathbb{Q}}$, we use a theorem of Jakhar, Khanduja and Sangwan to give explicit descriptions of all monogenic trinomials $f(x)$ having Galois group $G$. We also determine when these descriptions provide infinitely many such trinomials, and we investigate when these trinomials generate distinct sextic fields. These results extend recent work on monogenic power-compositional sextic trinomials of the form $g(x^3)$ to the situation $g(x^2)$, and thereby complete the characterization, in terms of their Galois groups, of monogenic power-compositional sextic trinomials. (English) |
| Keyword:
|
monogenic |
| Keyword:
|
even sextic |
| Keyword:
|
trinomial |
| Keyword:
|
Galois |
| Keyword:
|
power-compositional |
| MSC:
|
11R09 |
| MSC:
|
11R21 |
| MSC:
|
11R32 |
| DOI:
|
10.5817/AM2026-1-19 |
| . |
| Date available:
|
2026-04-23T07:59:46Z |
| Last updated:
|
2026-04-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153604 |
| . |
| Reference:
|
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