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Multiplicative independence in function fields. (English) Zbl 0780.11058
Let \(F\) be a field of characteristic \(p>0\) and suppose that \(k\) is a function field in one variable over \(F\) (i.e., suppose that there exists an element \(t\in k\) transcendental over \(F\) such that \(k\) is a finite extension of \(F(t)\)). Suppose that \(\nu\) is a discrete rank 1 valuation on \(k\) which is trivial on \(F\), and denote by \(\widehat {k}\) the completion of \(k\) with respect to \(\nu\). Let \(U_ 1\subset \widehat {k}^ \times\) denote the multiplicative group of 1-units of \(\widehat{k}\). Then \(U_ 1\) is a module over the ring of \(p\)-adic integers \(\mathbb{Z}_ p\) in a natural way. Call \(u_ 1,u_ 2,\dots,u_ n\in U_ 1\) multiplicatively independent over \(\mathbb{Z}_ p\) if the relation \(u_ 1^{\alpha_ 1} u_ 2^{\alpha_ 2} \dots u_ n^{\alpha_ n}=1\) with \(\alpha_ 1,\alpha_ 2,\dots,\alpha_ n\in \mathbb{Z}_ p\) implies that \(\alpha_ 1= \alpha_ 2= \dots= \alpha_ n=0\).
Consider the case that \(F\) is a perfect field, and denote by \(F_ 0\) the residue class field of \(\nu\). Then \(F_ 0/F\) is a finite separable extension and \(\widehat {k}\simeq F_ 0((\pi))\) where \(\pi\) is a uniformizing parameter for \(\nu\). Denote by \(\overline{k}\) the algebraic closure of \(k\) in \(\widehat {k}\). The author proves a result which is an analogue for function fields of a strong form of Leopoldt’s conjecture for number fields.
Every set of 1-units (with respect to \(\nu\)) of \(k\) which is multiplicatively independent over \(\mathbb{Z}\) is also multiplicatively independent over \(\mathbb{Z}_ p\) in \(U_ 1\).

11R58 Arithmetic theory of algebraic function fields
12F10 Separable extensions, Galois theory
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