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On multiplicative and linear independence of polynomial roots. (English) Zbl 0757.12001

General algebra, Proc. Conf., Vienna/Austria 1990, Contrib. Gen. Algebra 7, 127-135 (1991).
[For the entire collection see Zbl 0731.00007.]
The paper studies multiplicative and additive independence relations for the roots of polynomials over fields. In the multiplicative case holds: Let \(p\) be an odd prime number, \(K\) a field, \(f(x)\in K[x]\) an irreducible polynomial \(\neq x^ p+a_ 0\) of degree \(p\) over \(K\) without multiple roots, and \(x_ 1,x_ 2,\dots,x_ n\in\overline K\) its roots. Furthermore, let the degree of the extension \(K(\zeta_ m)\) : \(K\) be different from \(p\) for all positive integers \(m\), where \(\zeta_ m\) denotes a primitive \(m\)-th root of unity. Then the product \(\prod_{i=1}^ p x_ i^{k_ i}\in K\) for arbitrary integers \(k_ 1,k_ 2,\dots,k_ p\) if and only if \(k_ 1=k_ 2=\dots+k_ p\).
In the additive case the authors prove: Let \(p\) be an odd prime, \(K\) a field with char\((K)\neq p\), \(f(x)\in K[x]\) an irreducible polynomials of degree \(p\) over \(K\) without multiple roots, and \(x_ 1,x_ 2,\dots,x_ p\in\overline K\) its roots. Furthermore, assume that \((K(\zeta_ p):K)=p-1\), where \(\zeta_ p\) denotes a primitive \(p\)-th root of unity. Then the sum \(\sum_{i=1}^ p k_ i x_ i\in K\) for arbitrary \(k_ 1,k_ 2,\dots,k_ p\in K\) if and only if \(k_ 1=k_ 2=\dots=k_ p\).

MSC:

12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
12E05 Polynomials in general fields (irreducibility, etc.)

Citations:

Zbl 0731.00007
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