Abstract
Let <italic>E/F</italic> be a Galois extension of number fields with Galois group <italic>G</italic>=Gal(<italic>E/F</italic>), and let <italic>p</italic> be a prime not dividing #<italic>G</italic>. In this paper, using character theory of finite groups, we obtain the upper bound of #<italic>K</italic><sub>2</sub><italic>O<sub>E</sub></italic> if the group <italic>K</italic><sub>2</sub><italic>O<sub>E</sub></italic> is cyclic, and prove some results on the divisibility of the <italic>p</italic>-rank of the tame kernel <italic>K</italic><sub>2</sub><italic>O<sub>E</sub></italic>, where <italic>E/F</italic> is not necessarily abelian. In particular, in the case of <italic>G</italic>=<italic>C<sub>n</sub>, D<sub>n</sub>, A</italic><sub>4</sub>, we easily get some results on the divisibility of the <italic>p</italic>-rank of the tame kernel <italic>K</italic><sub>2</sub><italic>O<sub>E</sub></italic> by the character table. Let <italic>E</italic>/Q be a normal extension with Galois group <italic>D<sub>l</sub></italic>, where <italic>l</italic> is an odd prime, and <italic>F</italic>/Q a non-normal subextension with degree <italic>l</italic>. As an application, we show that <italic>f</italic>|<italic>p</italic>-rank <italic>K</italic><sub>2</sub><italic>O<sub>F</sub></italic>, where <italic>f</italic> is the smallest positive integer such that <italic>p<sup>f</sup></italic>≡±1(mod <italic>l</italic>).
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