论文标题
关于nilpotent内核有限嵌入问题的注释
A note on finite embedding problems with nilpotent kernel
论文作者
论文摘要
本说明的第一个目的是通过证明全球字段$ k $和有限的$ k $ primes的有限套装$ k $ $ k $ $ k $的$ k $ $ k $ $ k $ $ g \ rightarrow {。 $ {\ rm {gal}}（f/k）\ rightArrow g $，以使$ \ MATHCAL {S} $中的所有总理完全分为$ f/l $。然后，我们将其应用于分层环的逆向加洛伊斯理论。首先，考虑到一个数字字段$ k $的水平至少$ 4 $，我们表明每个有限解决的组都是一个galois组，而在$ k $中具有系数的分区环$ h_k $ h_k $ h_k $ h_k $。其次，鉴于有限的nilpotent内核在有限的字段$ k $上存在有限的拆分嵌入问题，我们充分描述了$ k $的自动形态$σ$ $ k $ $ k $的嵌入问题在分数$ k（t，σ）$ k（t，σ）$ k（t，σ）$ k（t，σ）上获得了解决方案。
The first aim of this note is to fill a gap in the literature by proving that, given a global field $K$ and a finite set $\mathcal{S}$ of primes of $K$, every finite split embedding problem $G \rightarrow {\rm{Gal}}(L/K)$ over $K$ with nilpotent kernel has a solution ${\rm{Gal}}(F/K) \rightarrow G$ such that all primes in $\mathcal{S}$ are totally split in $F/L$. We then apply this to inverse Galois theory over division rings. Firstly, given a number field $K$ of level at least $4$, we show that every finite solvable group occurs as a Galois group over the division ring $H_K$ of quaternions with coefficients in $K$. Secondly, given a finite split embedding problem with nilpotent kernel over a finite field $K$, we fully describe for which automorphisms $σ$ of $K$ the embedding problem acquires a solution over the skew field of fractions $K(T, σ)$ of the twisted polynomial ring $K[T, σ]$.
