On the Galois group of a reciprocal even octic polynomial. (English) Zbl 07841625
An ongoing research problem in Galois theory is to determine conditions for \(\mathrm{Gal}(f)\), the Galois group of a given polynomial \(f(x)\). Of recent interest are irreducible polynomials of the form \(f(x)=g(x^k)\) for some integer \(k \ge 2\).
Let \(F\) be a field of characteristic zero. Complete characterisations of \(\mathrm{Gal}(f)\) using only the coefficients of \(f(x) \in F[x]\) have been determined for the cases where
The paper under review extends the known literature by considering the case where \(k=2\) and \(g(x)=x^4+ax^3+bx^2+ax+1\). The authors showed that \(\mathrm{Gal}(f)\) identified as a transitive subgroup of \(S_8\) up to conjugacy belongs to one of six possible groups. By a detailed analysis of three distinct quartic subfields defined by \(f(x)\), they provid elementary characterisations for each possible group that only involves testing whether few elements in \(F\) obtained from the coefficients of \(f(x)\) are perfect squares. As an application, the authors give one-parameter families of polynomials for each of the six possible Galois groups.
Let \(F\) be a field of characteristic zero. Complete characterisations of \(\mathrm{Gal}(f)\) using only the coefficients of \(f(x) \in F[x]\) have been determined for the cases where
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- \(k=2\) and \(g(x)=x^2+ax+b\). [L.-C. Kappe and B. Warren, Am. Math. Mon. 96, No. 2, 133–137 (1989; Zbl 0702.11075)]
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- \(k=3\) and \(g(x)=x^2+ax+b\). [C. Awtrey and P. Jakes, Can. Math. Bull. 63, No. 3, 670–676 (2020; Zbl 1458.11173)]
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- \(k=2\) and \(g(x)=x^3+ax^2+bx+c\). [C. Awtrey, J. R. Beuerle and H. N. Griesbach, Missouri J. Math. Sci. 33, No. 2, 163–180 (2021; Zbl 1486.12004)]
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- \(k=2\) and \(g(x)=x^4+ax^2+b\). [C. Awtrey and F. Patane, J. Algebra Appl. (to appear)]
The paper under review extends the known literature by considering the case where \(k=2\) and \(g(x)=x^4+ax^3+bx^2+ax+1\). The authors showed that \(\mathrm{Gal}(f)\) identified as a transitive subgroup of \(S_8\) up to conjugacy belongs to one of six possible groups. By a detailed analysis of three distinct quartic subfields defined by \(f(x)\), they provid elementary characterisations for each possible group that only involves testing whether few elements in \(F\) obtained from the coefficients of \(f(x)\) are perfect squares. As an application, the authors give one-parameter families of polynomials for each of the six possible Galois groups.
Reviewer: Malcolm Hoong Wai Chen (Kuala Lumpur)
MSC:
| 12F10 | Separable extensions, Galois theory |
| 11Y40 | Algebraic number theory computations |
| 12-08 | Computational methods for problems pertaining to field theory |
Software:
GAPReferences:
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| [2] | Awtrey, C., Jakes, P. (2020). Galois groups of even sextic polynomials. Can. Math. Bull.63(3):670-676. DOI: . · Zbl 1458.11173 |
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| [9] | Ide, J., Jones, L. (2014). Infinite families of A_4-sextic polynomials. Can. Math. Bull.57(3):538-545. DOI: . · Zbl 1372.12001 |
| [10] | Jones, L. (2022). Monogenic reciprocal trinomials and their Galois groups. J. Algebra Appl.21(2):Paper No. 2250026, 11. DOI: . · Zbl 1500.11075 |
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