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In this paper we study the structure of the $3-$part of the ideal class group of a certain family of real cyclotomic fields with $3-$class number exactly $9$ and conductor equal to the product of two distinct odd primes. We employ known results from Class Field Theory as well as theoretical and numerical results on real cyclic sextic fields, and we show that the $3-$part of the ideal class group of such cyclotomic fields must be cyclic. We present four examples of fields that fall into our category, namely the fields of conductor $3 \cdot 331$, $7 \cdot 67$, $3 \cdot 643$ and $7 \cdot 257$, and they are the only ones amongst all real cyclotomic fields with conductor $pq \leq 2021$. The $3-$part of the class number for the two fields of conductor $3 \cdot 643$ and $7 \cdot 257$ was up to now unknown and we compute it in this paper.

The 3-part of the ideal class group of a certain family of real cyclotomic fields