On Kummer Theory and the Number of Roots of Unity in Radical Extensions of Q

Johannes Blömer International Computer Science Institute Berkeley, USA e-mail: bloemer@ICSI.Berkeley.Edu Report B 93-14 August 93

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{\it Kummer theory} states that if $F$ is a field containing a primitive $d$-th root of unity then the subfields of a radical extension of $F$ generated by radicals $\sq[d]{\rh_{1}},\ldots,\sq[d]{\rh_{k}}$ over $F$ can be described by subgroups of the group of $d$-th powers of elements in $F\backslash\{0\}.$ Building on work of Kneser in this paper we show that the same result can be obtained if $F$ satisfies weaker conditions. For example, it suffices that for each prime divisor $p$ of $d$ the field $F$ contains primitive a $p$-th root of unity. This result is used to prove that an extension $\Q(\sq[d_{1}]{n_{1}},\sq[d_{2}]{n_{2}},\ldots,\sq[d_{k}]{n_{k}},\ze_{m}),$ $\sq[d_{i}]{n_{i}}$ a real radical over $\Q$, $\ze_{m}$ a primitive $m$-th root of unity, contains at most $24m$ roots of unity. The bound is optimal.