Linear relations between roots of polynomials.(English)Zbl 0924.12002

Acta Arith. 89, No. 1, 53-96 (1999); corrigendum ibid. 110, No. 2, 203 (2003).
Let $$f$$ be a monic irreducible polynomial of degree $$n$$ over a field $$K$$ of characteristic $$0$$, and let $$x_{1},x_{2},...,x_{n}$$ be the roots of $$f$$ in some splitting field $$L:=K(x_{1},x_{2},...,x_{n}).$$ The author is interested in the possible existence of (nontrivial) additive or multiplicative relations between the roots of the forms: $$\sum_{i=1}^{n}a_{i}x_{i}=0$$ (with $$a_{i}\in K$$) or $$\prod_{i=1}^{n}x_{i}^{a_{i}}=1$$ (with $$a_{i}\in\mathbb{Z}$$) where not all the $$a_{i}$$ are equal.
Consideration of problems of this type date back at least to the paper of the author [K. Girstmair, Manuscr. Math. 39, 81-97 (1982; Zbl 0514.12010)] and C. J. Smyth [J. Number Theory 23, 243-254 (1986; Zbl 0586.12001)]. The former of these papers seems to have been overlooked by most of the later writers. The kinds of possible relations depend strongly on the Galois group $$G:=Gal(L/K)$$ (see, for example, J. D. Dixon [Acta Arith. 82, 293-302 (1997; Zbl 0881.12001)]) but also on the specific polynomial; for example, two polynomials may have the same Galois group over $$K$$ where one has nontrivial linear relations and the other has none. The paper of Smyth and most recent papers on this topic have sought necessary conditions for a given relation to hold (usually one involving only a few roots).
In contrast, in the present paper the author starts with a Galois group $$G$$ and then looks for relations of the above type which can occur for some $$f$$ whose Galois group is $$G.$$ We describe his approach for additive relations (the approach for multiplicative relations is similar). Fix a root $$x$$ of $$f,$$ and define $$H:=G_{x}$$ as the subgroup of $$G$$ fixing $$x$$; this characterizes the permutation action of $$G$$ on the set of roots of $$f.$$ The $$K$$-module $$K[G/H]$$ whose basis is the set of left $$H$$-cosets in $$G$$ is a left $$K[G]$$-module in a natural way. Each linear combination $$\sum_{i=1}^{n}a_{i}x_{i}$$ of roots of $$f$$ may be written in a unique way in the form $$\alpha x$$ where $$\alpha\in K[G/H],$$ so we can identify relations for $$f$$ with elements of $$K[G/H].$$ Conversely, a subset $$M$$ of $$K[G/H]$$ is called admissible if its elements could correspond to relations for some irreducible polynomial $$f$$ over $$K$$ with a root $$x$$ in this way. Define the idempotent $$\varepsilon_{H}:=(\sum_{u\in H}u)/\left| H\right|$$ in $$K[G]$$. Then the essential criterion proved by the author (Theorem 1) can be expressed as follows.
A necessary and sufficient condition for a $$K[G]$$-submodule $$V$$ of $$K[G/H]$$ to be admissible is that whenever $$z\in G$$ and $$(z-1)\varepsilon_{H}\in V$$ then $$z\in H$$.
He now uses this criterion and representation theory for $$G$$ to analyse the possible relations which might hold for some polynomial with Galois group $$G$$ with a permutation action defined by a subgroup $$H.$$ It turns out that if this permutation representation is multiplicity-free as a $$K$$-linear representation, then the set of all possible relations can, in principle, be completely described (the “tame” case), but if the representation is not multiplicity-free there is no simple description (the “wild” case). As examples, the author computes all possible relations for the group $$G:=PSL(2,11)$$ in its permutation representation of degree 12, and shows that an abelian group $$G$$ (in its regular representation) has $$x_{1}=x_{2}+x_{3}$$ as a possible relation if and only if $$6\mid\left| G\right| .$$
Reviewer: J.D.Dixon (Ottawa)

MSC:

 12F10 Separable extensions, Galois theory 12E05 Polynomials in general fields (irreducibility, etc.)
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