论文标题
$ p $ adadic多项式的密度生成固定分裂类型的扩展
Density of $p$-adic polynomials generating extensions with fixed splitting type
论文作者
论文摘要
我们证明，多项式的密度$ p（x）= \ sum_ {i = 0}^n a_n x^n $在本地字段上$ k $生成具有指定分裂类型的étale扩展名，在拆分类型的情况下，$ k $的残基大小的大小是有理功能的。此外，我们为这些密度提供了可计算的递归公式，并计算出该密度的渐近物，因为残基场的大小倾向于无穷大。
We prove that the density of polynomials $P(x)=\sum_{i=0}^n a_n x^n$ over a local field $K$ generating an étale extension with specified splitting type is a rational function in terms of the size of the residue field of $K$ in the case where the splitting type is tame. Moreover, we give a computable recursive formula for these densities and compute the asymptotics of this density as the size of the residue field tends to infinity.
