论文标题
具有简单理论的大型田野的加洛伊斯（Galois）小组（菲利普·迪特曼（Philip Dittmann）的附录）
Galois groups of large fields with simple theory (with an appendix by Philip Dittmann)
论文作者
论文摘要
假设$ k $是一个无限的领域，它很大（从流行意义上），其一阶理论很简单。我们表明$ k $是{\ em限制}，即只有任何给定有限程度的可分离扩展。我们还表明，$ k $上的任何$ 0 $ curve都具有$ k $ - 点，如果$ k $是完美的，则$ k $具有微不足道的brauer group。这些结果为猜想提供了证据，表明大型简单领域是有界PAC的。将我们的结果与Lubotzky和van den Dries的定理相结合，我们表明有一个有限的$ \ mathrm {pac} $ field $ l $，与$ k $相同的绝对Galois Group。在附录中，我们表明，如果$ k $很大，而$ \ mathrm {nsop} _ \ infty $和$ v $是$ k $ the $ k $ then $（k，v）$的非客气估值，则可以分开封闭的亨氏化，因此，尤其是$（k，v）$的残基领域，$是Algebraionally Calue Group and algebraips nater Value Group divis difis difis difis difis difis divis difis difis difis difis difis divis。附录还表明，正式和正式的$ p $ - adiC字段为$ \ mathrm {sop} _ \ infty $（不假定宽敞）。
Suppose that $K$ is an infinite field which is large (in the sense of Pop) and whose first order theory is simple. We show that $K$ is {\em bounded}, namely has only finitely many separable extensions of any given finite degree. We also show that any genus $0$ curve over $K$ has a $K$-point and if $K$ is additionally perfect then $K$ has trivial Brauer group. These results give evidence towards the conjecture that large simple fields are bounded PAC. Combining our results with a theorem of Lubotzky and van den Dries we show that there is a bounded $\mathrm{PAC}$ field $L$ with the same absolute Galois group as $K$. In the appendix we show that if $K$ is large and $\mathrm{NSOP}_\infty$ and $v$ is a non-trivial valuation on $K$ then $(K,v)$ has separably closed Henselization, so in particular the residue field of $(K,v)$ is algebraically closed and the value group is divisible. The appendix also shows that formally real and formally $p$-adic fields are $\mathrm{SOP}_\infty$ (without assuming largeness).
