在基域扩张下的分解群和惰性群

Decomposition Groups and Inertia Groups under Extensions of the Basic Field

K_1和K_2是代数数域K_0上的有限Galois扩张,K是K_1和K_2的合成域,P和P_1分别是K和K_1中的素理想,PP_1,P对于K_2和P_1对于K_0的分解群分别记为Z_(1′)和Z_1,P对K_2和P_1对K_0的惰性群分别记为T_(1′)和T_1。我们证明了Z_(1′)和T_(1′),分别是Z_1和T_1的正规子群,得到了用Z_(1′)表示Z_1的一种方法、把T_(1′)扩成T_1的一种方法,Z_(1′)=Z_1的充要条件和T_(1′)=T_1的充要条件。

Suppose that K_1 and K_2 are both finite Galois extensions of a algebraic numberfield K_0;K=K_1K_2 is their composite field;p and p_1 are prime ideals of K andK_1 respectively,and PP_1;the decomposition groups of P with respect to K_2and P_1 with respect to K_0 are denoted by Z_1′ and Z_1 respectively;the inertiagroups of P with respect to K_2 and P_1 with respect to K_0 are denoted by T_1′ andT_1 respectively.We have proved that Z_1′ is a normal subgroup of Z_1 and T_1′ isa normal subgroup of T_1.We have obtained a method of expressing Z_1 in ferms ofZ_1′ and a method of extending T_1′ to T_1.We have also obtained the sufficientand necessary conditions for Z_1=Z_1′ and T_1=T_1′.The results in this paper arealso valid under non-Archinedean valuation field instead of a algebraic numberfield.