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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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