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Irreducible polynomials with many roots of maximal modulus. (English) Zbl 0813.11060

Let \(f(x)\) be an irreducible polynomial with integer coefficients. Suppose that \(f(x)\) has \(m\) roots of maximal modulus, at least one of them being real. Then there is a polynomial \(g\) with integer coefficients such that \(f(x) = g(x^ m)\). That is, the roots of largest modulus are exactly \(\alpha \omega^ j\), \(j = 0, \dots, m - 1\), where \(\omega\) is a primitive \(m\)-th root of unity, and \(\alpha\) is the real root of maximal modulus. The result is also true if “maximal modulus” is replaced by “minimal modulus”. The proof uses the transitivity of the Galois group of \(f(x)\) to show that the set of roots of maximal modulus is closed under the unusual operation \((a,b) \to a^ 2 b^{-1}\). Then an elementary argument completes the proof.

MSC:

11R09 Polynomials (irreducibility, etc.)
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