关于限制的Hilbert-Speiser和Leopoldt性质
On the restricted Hilbert-Speiser and Leopoldt properties
摘要：
设G是有限阿贝尔群.数域K称为G型Hilbert-Speiser域，如果对每个Tame G-Galois扩张L/K，整数环O_L作为O_K[G]-模是自由的.若O_L对每个G-Galois扩张L/K在相应阶A_{L/K}上是自由的，则称K为G型Leopoldt域.众所周知（并且很容易看出），如果K是G型的Leopoldt，则K是G型的Hilbert-Speiser。我们证明了，当G=C_p具有素数阶时，一般情况下，逆命题不成立，但对许多数域K（特别是对K/QGalois），一个修正的命题成立.我们给出G=C_p的例子来说明即使是修改的逆一般也是假的，并且当原逆不成立时，修改的逆也能成立。
英文摘要：
Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if, for every tame G-Galois extension L/K, the ring of integers O_L is free as an O_K[G]-module. If O_L is free over the associated order A_{L/K} for every G-Galois extension L/K, then K is called a Leopoldt field of type G. It is well-known (and easy to see) that if K is Leopoldt of type G, then K is Hilbert-Speiser of type G. We show that the converse does not hold in general, but that a modified version does hold for many number fields K (in particular, for K/Q Galois) when G=C_p has prime order. We give examples with G=C_p to show that even the modified converse is false in general, and that the modified converse can hold when the original does not.
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