Some polynomials over $\mathbb{Q}(t)$ and their Galois groups

Gene Ward Smith

Type: Article

Publication Date: 1999-08-19

Citations: 13

DOI: https://doi.org/10.1090/s0025-5718-99-01160-6

Abstract

Examples of polynomials with Galois group over $\mathbb Q(t)$ corresponding to every transitive group through degree eight are calculated, constructively demonstrating the existence of an infinity of extensions with each Galois group over $\mathbb Q$ through degree eight. The methods used, which for the most part have not appeared in print, are briefly discussed.

Locations

Mathematics of Computation - View - PDF

Similar Works

Action	Title	Year	Authors
+	Galois extensions of Q(T)	2007
+	Polynomials with galois group PSL(2,11)	1981	Samuel LaMacchia
+	Generic Polynomials for Q8-, QC-, and QQ-Extensions	2001	Arne Ledet
+	Polynomials with galois group psl(2,7)	1980	Samuel LaMacchia
+ PDF Chat	Generic polynomials over $\boldsymbol {Q}$ with two parameters for the transitive groups of degree five	2003	Ki-ichiro Hashimoto Hiroshi Tsunogai
+ PDF Chat	Linear relations among Galois conjugates over $${\mathbb {F}}_q(t)$$	2022	Will Hardt John Yin
+	A parametric family of quintic polynomials with Galois group D5	1982	G. Roland Noriko Yui Don Zagier
+ PDF Chat	Elliptic curve involving subfamilies of rank at least 5 over $\mathbb{Q}(t)$ or $\mathbb{Q}(t,k)$	2021	Ahmed El Amine Youmbai A. Muhammed Uludaǧ Djilali Behloul
+	A family of sextic polynomials with Galois group A5 : computation ofsplitting fields and Galois groups	1997	Hirokazu Anai
+	Galois extensions of C(T)	2007
+ PDF Chat	Polynomials with Galois groups 𝐴𝑢𝑡(𝑀₂₂),𝑀₂₂, and 𝑃𝑆𝐿₃(𝐹₄)⋅2₂ over 𝑄	1988	Gunter Malle
+	Polynomials with a given Galois group	1989	Augusto Nóbile
+	A Note on a General Class of Polynomials	1994	Richard André-Jeannin
+	Extensions of the Field of Rational Numbers Whose Galois Group IS Isomorphic To A Group of Order Eight	2018	Stephen Murithi Kimathi
+ PDF Chat	Prime specialization and iterates of composition of Polynomials	2021	Sushma Palimar
+	Field Extension by Galois Theory	2017	Md Taufiq Nasseef
+	Galois Theory: Fifth-Degree Polynomials and Some Ruler-and-Compass Problems	2020	Gabriel Joseph Loehr
+	An Exploration of Galois Theory with some Classical Results	2020	Olof Klingberg
+	Realisierung der Mathieugruppen M11 und M12 als Galoisgruppen über Q	1986	B. Heinrich Matzat Andreas Zeh-Marschke
+	Galois Theory	2002	Serge Lang

Works That Cite This (10)

Action	Title	Year	Authors
+	Division Algebras with PSL(2,q)-Galois Maximal Subfields	2001	Elizabeth S. Allman Murray Schacher
+ PDF Chat	Infinite Families of A<sub>4</sub>-Sextic Polynomials	2014	Joshua Ide Lenny Jones
+ PDF Chat	On the Common Index Divisors of a Dihedral Field of Prime Degree	2007	Blair K. Spearman Kenneth S. Williams Qiduan Yang
+	Enumerative Galois theory for cubics and quartics	2020	Sam Chow Rainer Dietmann
+	Galois Groups of Even Sextic Polynomials	2019	Chad Awtrey Peter Jakes
+	On monic abelian cubics	2022	Stanley Yao Xiao
+	Imprimitive octic fields with small discriminants	1998	Henri Cohen Francisco Diaz y Diaz Michel Olivier
+	Tables of octic fields with a quartic subfield	1999	Henri Cohen Fredéric Diaz Michel Olivier
+	Automorphic L-functions and their applications to Number Theory	2012	Jae‐Hyun Cho
+	Fortschritte in der inversen Galoistheorie	1999	B. Heinrich Matzat

Works Cited by This (10)

Action	Title	Year	Authors
+	Explicit construction of Ãn type fields	1989	Teresa Crespo
+ PDF Chat	On the construction of Galois extensions of function fields and number fields	1974	Kuang-yen Shih
+	The construction of SL(2, 3)-polynomials	1984	Franz-Peter Heider Paulgerd Kolvenbach
+	The transitive groups of degree up to eleven<sup>+</sup>	1983	Gregory Butler John McKay
+	Polynome mit Galoisgruppen<i>PGL</i><sub>2</sub>(p) und<i>PSL</i><sub>2</sub>(p) über	1993	Gunter Malle
+	Generic cyclic polynomials of odd degree	1991	Ward Smith Gene
+ PDF Chat	Computation of Galois groups over function fields	1997	Thomas W. Mattman John McKay
+	Polynomials with Cyclic Galois Group	1995	Ralf Dentzer
+ PDF Chat	On Transitive Permutation Groups	1998	John H. Conway Alexander Hulpke John McKay
+ PDF Chat	Konstruktive Galoistheorie	1987	B. Heinrich Matzat