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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 $$U_ 1$$.

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