Polynomial relations between polynomial roots
Let \(f\) be a separable polynomial of degree \(n\) over a field \(K\) and let \(G\) be the Galois group of its splitting field. It has been shown by \textit{C. J. Smyth} [J. Number Theory 23, 243-254 (1986; Zbl 0586.12001)]\ that if \(G= S_n\) and the characteristic of \(K\) either is zero or exceeds \(n\), then there are no non-trivial additive relations between roots of \(f\). The authors first extend this result to cover the case when \(G\) is doubly transitive and the characteristic of \(K\) is either zero or prime to \(|G|\). If \(G\) is \(2k\)-transitive for some \(k\) (which in case \(k\geq 3\) means that \(G\) equals either \(S_n\) or \(A_n\) and in case \(k=2\) it may be also equal to one of the Mathieu groups) then the authors show under the same assumptions that the only polynomial relation between the roots which is linear in every variable must be a linear combination of the elementary symmetric functions with the possible exception of the case \(\text{char} (K)\leq k\). After presenting some examples of non- trivial relations between roots of a polynomial the question of additive and multiplicative relations between conjugates of algebraic numbers is studied in the language of group rings.
scientific article; zbMATH DE number 827378