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Class numbers of cyclotomic function fields. (English) Zbl 0989.11064

Let \(A={\mathbb{F}}_q[T]\) be the ring of polynomials over a finite field \({\mathbb{F}} _q\) of \(q\) elements. Let \(k = {\mathbb{F}}_q (T)\) be the rational function field over \({\mathbb{F}}_q\). For \(M\in A\) let \(k\big(\Lambda _M\big)\) be the \(M\)-th Carlitz-Hayes cyclotomic function field. The real subfield \(k\big(\Lambda _M\big)^+\) of \(k\big(\Lambda _M\big)\) is defined as the maximal subfield of \(k\big(\Lambda _M\big)\) such that \(1/T\) splits completely. Let \(h(M)\) and \(h^+(M)\) be the class numbers of \(k\big(\Lambda _M\big)\) and \(k\big(\Lambda _M\big)^+\) respectively. As in the number field case, we have that \(h^+(M)\mid h(M)\) and we define \(h^-(M)\) by the equation \(h^+(M) h^-(M) = h(M)\). Furthermore, in the function field case we have that if \(M\mid N\) then \(h^-(M)\mid h ^- (N)\) and \(h^+(M)\mid h ^+(N)\).
In the paper under review, the authors obtain upper bounds for \(h^- (M)\) and \(h^+(M)\) in the case \(M=P\) is an irreducible polynomial. As a corollary, it is proved that if \(\deg P =1\) or \(2\), then \(h^+(P) = 1\) and if \(\deg P=1\), \(h^-(P) =1\). Next, the case \(M=P^n\) is considered, \(P\) irreducible and \(n\geq 1\). In this case, \(h^-(M)\) can be given in terms of the determinant of a matrix defined in terms of polynomials prime to \(M\) of degree less than the degree of \(M\). There are given several examples using the above characterization. Using this result, it is also proved that if \(\ell\) is a rational prime and \(\ell ^c\) is the highest power of \(\ell\) dividing \(q-1\), then \(\ell ^{c(r-1)} \mid h^-(M)\) and so \((q-1) ^r \mid h^-(M)\), where \(r ={{|(A/M) ^{\ast}|} \over {q-1}}\). As a corollary, it is obtained that \(h^{-1}(P^n) = 1\) if and only if \(\deg P = 1\) and \(n=1\).
In the last section the authors find all \(M\in A\) with \(h^-(M)=1\) when \(q\) is odd.
Reviewer’s remark: In Proposition 5 the irreducible polynomials dividing \(M\) are of degree 1 and \(q\) is odd.

MSC:

11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.)
11R29 Class numbers, class groups, discriminants
11T55 Arithmetic theory of polynomial rings over finite fields
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