We prove the existence of the maximal real subfields of cyclotomic extensions over the rational function field $k={\mathbb{F}}_q(T)$ whose class groups can have arbitrarily large ${\ell }^n$-rank, where ${\mathbb{F}}_q$ is the finite field of prime power order q. We prove this in a constructive way: we explicitly construct infinite families of the maximal real subfields $k{(\mathrm{\Lambda })}^{+}$ of cyclotomic function fields $k(\mathrm{\Lambda })$ whose ideal class groups have arbitrary ${\ell }^n$-rank for n = 1, 2, and 3, where ℓ is a prime divisor of $q-1$. We also obtain a tower of cyclotomic function fields $K_i$ whose maximal real subfields have ideal class groups of ${\ell }^n$-ranks getting increased as the number of the finite places of k which are ramified in $K_i$ get increased for $i\ge 1$. Our main idea is to use the Kummer extensions over k which are subfields of $k{(\mathrm{\Lambda })}^{+}$, where the infinite prime ∞ of k splits completely. In fact, we construct the maximal real subfields $k{(\mathrm{\Lambda })}^{+}$ of cyclotomic function fields whose class groups contain the class groups of our Kummer extensions over k. We demonstrate our results by presenting some examples calculated by MAGMA at the end.

我们证明了有理函数场上最大的环原子扩展实子场的存在 $ķ={\mathbb{F}}_q（Ť）$班级人数可以任意大 ${\ell }^{ñ}$-等级，在哪里 ${\mathbb{F}}_q$是素数次幂q的有限域。我们以建设性的方式证明了这一点：我们显式构造了最大实子域的无限族$ķ{（\mathrm{\Lambda }）}^{+}$ 功能的领域 $ķ（\mathrm{\Lambda }）$ 理想的阶级群体具有任意性 ${\ell }^{ñ}$秩为Ñ = 1，2，和3，其中ℓ是的素因子$q-\mathrm{1个}$。我们还获得了一个具有连锁功能场的塔${ķ}_{\mathrm{一世}}$ 其最大实子字段具有理想的类别组 ${\ell }^{ñ}$-ranks越来越增大的有限位数ķ其在网状${ķ}_{\mathrm{一世}}$ 得到增加 $\mathrm{一世}\ge \mathrm{1个}$。我们的主要思想是使用k的Kummer扩展，它是$ķ{（\mathrm{\Lambda }）}^{+}$，其中k的无限质数∞完全分裂。实际上，我们构造了最大的实子域$ķ{（\mathrm{\Lambda }）}^{+}$环函数字段的类组包含我们在k上的Kummer扩展的类组。最后，我们将通过MAGMA计算得出的一些示例来证明我们的结果。