The universal thickening of the eld of real numbers
Alain Connes, Caterina Consani · arXiv (Cornell University) · 2012
We dene the universal 1-adic thickening of the eld of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process. We show that the transposition of the perfection process at the real archimedean place is identical to the \dequantization process and yields Viro’s tropical real hypereld T R. Then we prove that the archimedean Witt construction in the context of hyperelds allows one to recover a eld from a hypereld, and we obtain the universal pro-innitesimal thickening R1 of R. Finally, we provide the real analogues of several algebras used in the construction of the rings of p-adic periods. We