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On the subfields of cyclotomic function fields. (English) Zbl 1290.11155

Let \(K={\mathbb F}_q(T)\) be the rational function field over the finite field with \(q\) elements, where \(q\) ia a power of an odd prime. Let \(P\) be a monic irreducible polynomial in \({\mathcal O}_K={\mathbb F}_q[T]\) of degree \(d>0\), \(K_P=K(\lambda _P)\) be the cyclotomic function field where \(\Lambda _P\) is the set of \(P\)-torsion elements in the Carlitz \({\mathcal O}_K\)-module \(\bar K\) (i.e., the algebraic closure of \(K\)) and \(K^{+}_P\) be the maximal real subfield of \(K_P\).
It is well-known that \(K_P/K\) and \(K_P^{+}/K\) are cyclic extensions, so they contain unique subextensions \(M\) and \(M^{+}\) of degree \(k\) for each \(k\) dividing their order. In the case where \(k\mid q-1\), a formula for the analytic class number of \(M\) and \(M^{+}\) is given.

MSC:

11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.)
11R29 Class numbers, class groups, discriminants
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References:

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