6533b837fe1ef96bd12a2662

RESEARCH PRODUCT

Zu einem Satz von Isaacs �ber das Casus-Irreducibilis Ph�nomen

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Let \(\Omega \) be a field (of characteristic 0). A prime p is called “bose” (naughty) if \(\Omega \) contains all p-th roots of unity. In this paper the theorem is proved: Let K be an admissible subfield of \(\Omega \) (i.e. for each prime p K contains all p-th roots of unity lying in \(\Omega \)), a an algebraic element of \(\Omega /K\) which is contained in a repeated radical extension of K lying in \(\Omega \). Furthermore let the normal hull L of a over K be contained in \(\Omega \). Then all prime divisors of \(\mid L : K \mid \) are naughty (and L is a repeated radical extension of K with naughty prime exponents). This result generalises a theorem of Isaacs [1] who treats the case \(\Omega = {\Bbb R}\).