Özet:

Let $F$ be a global function field over the finite constant field $\mathbb{F}_q$ with $3\mid q-1$, and let $K/F$ be a cubic cyclic function fields extension with Galois group $G=$Gal$(K/F)=$. Denote by $\mathcal{C}(K)$ and $\mathcal{C}(K)_3$ the ideal class group of $K$ and its Sylow 3-subgroup, respectively. Let $\mathcal{C}(K)_3^G=\{[\fa]\in \mathcal{C}(K)_3|\ \sigma[\fa]=[\fa]\}$ and $\mathcal{C}(K)_3^{1-\sigma}=\{[\fa](\sigma[\fa])^{-1}|\ [\fa]\in \mathcal{C}(K)_3\}$. In this paper, we present a method for computing the 3-rank of the quotient group $\mathcal{C}(K)_3^G\mathcal{C}(K)_3^{1-\sigma}/\mathcal{C}(K)_3^{1-\sigma}$. Specifically, when $K$ is a cubic Kummer extension of $\mathbb{F}_q(T)$, we determine explicitly the key factors $t$, $x_1,\cdots, x_t$, and $[\mathfrak{A}_1],\cdots, [\mathfrak{A}_t]$ in the process of computing the 3-rank of $\mathcal{C}(K)_3^G\mathcal{C}(K)_3^{1-\sigma}/\mathcal{C}(K)_3^{1-\sigma}$. Combining this deterministic algorithm along with the structure of class groups for cubic Kummer function fields, the 3-rank of the Sylow 3-subgroup of $\mathcal{C}(K)$ is determined explicitly in this specific case. Examples are given in the last two sections to elucidate our computational method.

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