P次循环扩张F／Q

利用分圆域Q(ζ)的根式扩张,对有理数域的(奇素数)p次循环扩张进行了研究,得到了任意p次循环扩张的定义方程.文1],2]在p=3时已有了很好的结果;本文讨论了p=5的情形,最后得到一类由含一个有理参数的五次方程确定的五次循环扩张.

pth CYCLIC EXTENSIONS OVER Q

By making use of radical extensions over cyclotomic fields, we try to depict definition polynomials of cyclic extensions over Q of prime degree p and generalize the well-known correspondence between all quadratic fields and subgroups id Q/Q2 of order 2.Although the polynomials look somewhat "clumsy" since they involve complex combinatoral equations, the results in the cubic case which coincide with Prof. Li's are truly beautiful; and when p=5, we find the interestingTheorem. The splitting field is wildly ramified quintic cyclic over Q for all aeZ, where (x) is the quintie cyclotomic polynomial.