拟Eisenstein型数域中素数的分解

On The Decomposition of Rational Prime in the Pseudo-Eisenstein Field

Eisenstein型数域在素理想的分解研究中有着十分重要的作用。若将Eisenstein型数域进行推广,就会得到在更广泛的数域中素理想分解的信息。如果将代数整数ω的不可约多项式的条件减弱,就得到Eisenstein型数域的推广。本文尝试推广Eisenstein型数域为拟Eisenstein型数域K=(E,p,k),并且探讨在这样推广的条件下素理想分解的相应结果。利用Newton折线图,证明了在拟Eisenstein型数域(E,p,k)中素数p有e(P/p)=k的的素理想因子P,在k=n,n-1时,通过计算代数整数的范数证明了p在K中的分解满足Dedekind的引理,从而给出了素理想P的具体形式。对于拟Eisenstein域(E,p,k)的判别式中p的个数利用赋值方法做了估计,证明了pk-1整除判别式d(K)。

Eisenstein fields play an important role in the decomposition of rational prime. One of the most important conclusion says that, If ωis an algebraic number and its irreducible polynomial is p-Eisenstein, then p is completely ramifies in K= Q（ω）. On the other hand, algebraic fields such as Q（n√p） are Eisenstein fields. In the study of pure cubic fields, in many case we will found the problem can be solved in Eisenstein fields. If we generalized the Eisenstein fields, we will get much information of the decomposition of prime ideals. The method is loose the condition of the Eisenstein polynomial, acquire pseudo-Eisenstein fields. In this paper, we generalize the Eisenstein field and discuss the decomposition of rational prime in pseudo-Eisenstein fields. We find the rational prime p has a factor ideal P with e（P/p）=k by Newton polygon. When k=n,n-1 we calculate the norm of some algebraic number. We find the prime number p satisfies the condition of a theorem of Dedekind. Thus we can determine form of ideal P. We also determine the number of p in the discriminant of the pseudo-Eisenstein field （E,p,k）. We prove p^k-1 divides the discriminant d（K）.