## Units and class groups in cyclotomic function fields.(English)Zbl 0483.12003

Let $$p$$ be an odd prime. Let $$w$$ denote a primitive $$p$$-th root of 1. In the mid 19th century, Kummer proved that the class number of $$\mathbb Z[x + w^{-1}]$$ (the ring of integers of the maximal real subfield of $$\mathbb Z[w])$$ equals the index in $$\mathbb Z[w]^*$$ (the unit group of $$\mathbb Z[w])$$ of the subgroup of circular units. The latter is the subgroup of $$\mathbb Z[w]^*$$ generated by $$(1 - w^n)/( 1 - w)$$ for $$1\le n\le p-1$$. In 1978 this theorem was generalized by Sinnott to the case that $$w$$ is a primitive $$m$$-th root of 1 where $$m$$ is an arbitrary positive integer.
Let $$\mathbb F$$ be a field with $$q$$ elements and let $$k = \mathbb F(T)$$ be a rational function field in one variable over $$\mathbb F$$. For each monic polynomial $$M\in \mathbb F[T]$$, Carlitz associated an abelian extension $$k(M)$$ of $$k$$. The field $$k(M)$$ is analogous in many ways to the cyclotomic extension $$\mathbb Q(w)$$ of $$\mathbb Q$$.
In this paper the authors prove an analogue of Sinnott’s theorem for the “cyclotomic function field” $$k(M)$$. First, they define the “maximal real subfield” $$F(M)$$ of $$k(M)$$. They establish basic arithmetic properties and analytic class number formulas for the fields $$k(M)$$ and $$F(M)$$. Let $$\mathfrak O$$ denote the integral closure of $$\mathbb F[T]$$ in $$k(M)$$. The authors define a subgroup $$C$$ of $$\mathfrak O^*$$ (the units of $$\mathfrak O)$$ in analogy with the circular units in cyclotomic number fields. Let $$h$$ denote the class number of the integral closure of $$\mathbb F[T]$$ in $$f(M)$$ and let $$g$$ denote the number of monic prime factors of the polynomial $$M$$. The principal result is
$[\mathfrak O^* : C] = h\cdot (q - 1)^a$
where $$a = 0$$ if $$g = 1$$ and $$a = 2^{g-2}+1- g$$ if $$g>1$$. The proof of this formula involves the study of distributions on $$k$$ and an analysis of certain related cohomology groups.

### MSC:

 11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.) 11R18 Cyclotomic extensions 11R27 Units and factorization 11R29 Class numbers, class groups, discriminants
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### References:

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