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For any square-free positive integer $m\equiv 10(mod16)$ with $m\ge 26$, we prove that the class number of the real cyclotomic field $\mathbb{Q}({\zeta }_{4m}+{\zeta }_{4m}^{-1})$ is greater than $1$, where ${\zeta }_{4m}$ is a primitive $4m$th root of unity.

For any integer $n>1$, we provide a parametric family of biquadratic fields with class groups having $n$-rank at least 2. Moreover, in some cases, the $n$-rank is bigger than 4.