@article{92dbd09cfa0346a6ad3198824138a091,

title = "Reconstructing function fields from Milnor K-theory",

abstract = "Let F be a finitely generated regular field extension of transcendence degree ≥ 2 over a perfect field k. We show that the multiplicative group F ×/ k× endowed with the equivalence relation induced by algebraic dependence on F over k determines the isomorphism class of F in a functorial way. As a special case of this result, we obtain that the isomorphism class of the graded Milnor K-ring K M (F) determines the isomorphism class of F, when k is algebraically closed or finite.",

keywords = "Function fields, Milnor K-theory, Reconstruction",

author = "Anna Cadoret and Alena Pirutka",

note = "Funding Information: We express heartfelt thanks to the referees for their thorough and constructive reports. They pointed out several minor but mathematical inaccuracies and made suggestions that helped significantly improve the exposition. Cadoret was partially supported by the IUF and the ANR grant ANR-15-CE40-0002-01. Pirutka was partially supported by NSF grant DMS-1601680 and by the Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. no. 14.641.31.0001. This project was initiated at the occasion of a working group on geometric birational anabelian geometry at the Courant Institute of Mathematics, NYU. The authors would like to thank F. Bogomolov and Y. Tschinkel for inspiring exchanges, B. Kahn for his explanations concerning the Bass–Tate conjecture and A. Merkurjev, M. Morrow and A. Tamagawa for helpful remarks on a first version of this text. They also thank Tamagawa for suggesting the more elementary proof of Lemma 20 presented in Remark 22. Publisher Copyright: {\textcopyright} 2021 Mathematical Sciences Publishers.",

year = "2021",

doi = "10.2140/ant.2021.15.2261",

language = "English (US)",

volume = "15",

pages = "2261--2288",

journal = "Algebra and Number Theory",

issn = "1937-0652",

publisher = "Mathematical Sciences Publishers",

number = "9",

}