Does p divide the class number of the cubic field of conductor p?

Say p is 1 mod 3. Then there's a unique real cubic field in $\mathbb{Q}(\zeta_p)$. Does $p$ divide its class number? If no, this implies that the eigenspaces of the class group of $\mathbb{Q}(\zeta_p)$ corresponding to $(p-1)/3$ and $2(p-1)/3)$ vanish.