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Let $p$ be an odd prime, $r$ be a primitive root modulo $p$ and $r_i\equiv r^i(modp)$ with $1\le r_i\le p-1$. In 2007, R. Queme raised the question whether the $\ell$-rank ($\ell$ an odd prime $\ne p$) of the ideal class group of the $p$-th cyclotomic field is equal to the degree of the greatest common divisor over the finite field ${𝔽}_{\ell }$ of $x^{(p-1)/2}+1$ and Kummer’s polynomial $f\left(x\right)={\sum }_{i=0}^{p-2}{r}_{-i}{x}^i$. In this paper, we shall give the complete answer for this question enumerating a counter-example.