函数域上复合位的稠密性及其在实全纯环上的应用
Density of Composite Places in Function Fields and Applications to Real Holomorphy Rings
摘要：
给定一个代数函数域$F|K$和$K$上的一个库所$\wp$，我们证明了由$\wp$的扩张到$K$的有限扩张的复合库所在$F$的所有库所的空间中是强稠密的.我们把这个结果应用到$K=R$的情况，它是一个实闭域，并且在$R$上的不动位置是它的自然（最好）实位置。这导致了一个新的描述真正的全纯环的$F$可以被看作是一个模拟的某种改进阿丁的解决方案希尔伯特的第17个问题。我们还确定了$F$的所有$\R$ -位所构成的拓扑空间$M(F)$（剩余域包含在$\R$中的位所），与$R$的自然$\R$ -位所合成的$F$的所有$\R$ -位所构成的子空间，以及所有$R$ -有理位所构成的拓扑空间之间的关系.进一步证明了关于这些空间以及各类相对实全纯环的结果。在本文的最后，将应用环的实谱理论从这个角度解释基本概念，并表明空间$M(F)$只有1000个拓扑分量。
英文摘要：
Given an algebraic function field $F|K$ and a place $\wp$ on $K$, we prove that the places that are composite with extensions of $\wp$ to finite extensions of $K$ lie dense in the space of all places of $F$, in a strong sense. We apply the result to the case of $K=R$ any real closed field and the fixed place on $R$ being its natural (finest) real place. This leads to a new description of the real holomorphy ring of $F$ which can be seen as an analogue to a certain refinement of Artin's solution of Hilbert's 17th problem. We also determine the relation between the topological space $M(F)$ of all $\R$-places of $F$ (places with residue field contained in $\R$), its subspace of all $\R$-places of $F$ that are composite with the natural $\R$-place of $R$, and the topological space of all $R$-rational places. Further results about these spaces as well as various classes of relative real holomorphy rings are proven. At the conclusion of the paper the theory of real spectra of rings will be applied to interpret basic concepts from that angle and to show that the space $M(F)$ has only finitely many topological components.
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