Grunwald-Wang theorem, an effective version

The main purpose of this article is to establish an effective version of the Grunwald-Wang theorem, which asserts that given a family of local characters Xv of Kv of exponent m, where v C S for a finite set S of primes of K, there exists a global character X of the idele class group CK of exponent m （unless some special case occurs, when it is 2m） whose local component at v is Xv. The effectiveness problem for this theorem is to bound the norm N（X） of the conductor of X in terms of K, m, S and N（Xv） （v C S）. The Kummer case （when K contains pro） is easy since it is almost an application of the Chinese remainder theorem. In this paper, we solve this problem completely in general case, and give three versions of bound, one is with GRH, and the other two are unconditional bounds. These effective results have some interesting applications in concrete situations. To give a simple example, if we fix p and l, one gets a good least upper bound for N such that p is not an /-th power rood N. One also gets the least upper bound for N such that lr ｜φ｜（N） and p is not an/-th power mod N. Some part of this article is adopted （with some revision） from the unpublished thesis by Wang （2001）.

The main purpose of this article is to establish an effective version of the Grunwald-Wang theorem,which asserts that given a family of local characters χvof K *vof exponent m, where v ∈ S for a finite set S of primes of K, there exists a global character χ of the idele class group CK of exponent m(unless some special case occurs, when it is 2m) whose local component at v is χv. The effectiveness problem for this theorem is to bound the norm N(χ) of the conductor of χ in terms of K, m, S and N(χv)(v ∈ S). The Kummer case(when K contains μm) is easy since it is almost an application of the Chinese remainder theorem. In this paper, we solve this problem completely in general case, and give three versions of bound, one is with GRH, and the other two are unconditional bounds. These effective results have some interesting applications in concrete situations. To give a simple example, if we fix p and l, one gets a good least upper bound for N such that p is not an l-th power mod N. One also gets the least upper bound for N such that lr| φ(N) and p is not an l-th power mod N.Some part of this article is adopted(with some revision) from the unpublished thesis by Wang(2001).