#### Vol. 15, No. 9, 2021

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Reconstructing function fields from Milnor K-theory

### Anna Cadoret and Alena Pirutka

Vol. 15 (2021), No. 9, 2261–2288
##### Abstract

Let $F$ be a finitely generated regular field extension of transcendence degree $\ge 2$ 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 ${K}_{\ast }^{M}\left(F\right)$ determines the isomorphism class of $F$, when $k$ is algebraically closed or finite.

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