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On the simplest quartic fields and related Thue equations. (English) Zbl 1346.11029

Feng, Ruyong (ed.) et al., Computer mathematics. 9th Asian symposium, ASCM 2009, Fukuoka, Japan, December 14–17, 2009, 10th Asian symposium, ASCM 2012, Beijing, China, October 26–28, 2012. Contributed papers and invited talks. Berlin: Springer (ISBN 978-3-662-43798-8/hbk; 978-3-662-43799-5/ebook). 67-85 (2014).
Simplest cubic, simplest quartic and simplest sextic fields are families of totally real cyclic fields in which the Galois group is generated by a simple birational transformation \(x\rightarrow \frac{ax+b}{cx+d}\). Simplest quartic fields are generated over \(\mathbb Q\) by a root of the polynomial \[ x^4-mx^3-6x^2 + mx+1 \] where \(m\) is an integer parameter. These fields and the corresponding Thue equations \[ x^4-mx^3y-6x^2y^2 + mxy^3+y^4=c \] in \(x,y\in\mathbb Z\), have been studied by several authors.
Let \(K\) be a field of \(\text{char}\,K\neq 2\). For \(a\in K\), the author gives an explicit answer to the field isomorphism problem of the simplest quartic polynomial \(x^4- ax^3- 6x^2+ax + 1\) over K as the special case of the field intersection problem via multiresolvent polynomials. Over an infinite field \(K\) this result implies that the polynomial gives the same splitting field over \(K\) for infinitely many values \(a\) of \(K\).
For the entire collection see [Zbl 1309.68008].

MSC:

11D25 Cubic and quartic Diophantine equations
11D59 Thue-Mahler equations
11R16 Cubic and quartic extensions

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