By an algebraic number field, we mean a finite extension field of the field of rational numbers. Let $k$ be an algebraic number field, we denote by $\mathcal{O}_k$ the ring of algebraic integers in $k$. Let $K$ be a finite extension field of an algebraic number field $k$. Suppose for every ideal $I$ of $\mathcal{O}_k$, $I\mathcal{O}_K$ is principal. Then $K$ is called a PIT(Principal Ideal Theorem) field over $k$. Let $K$ be a PIT field over $k$. We say $K$ is a minimal PIT field over $k$ if $L/k$ is not PIT for every proper subextension $L/k$ of $K/k$.

(1) Let $k$ be an algebraic number field and $K/k$ be a finite extension. Is $K/k$ a minimal PIT if and only if $K/k$ is the Hilbert class field?

(2) Let $k$ be an algebraic number field and $K/k$ and $L/k$ be minimal PITs. Are $K/k$ and $L/k$ isomorphic?

By an algebraic number field, we mean a finite extension field of the field of rational numbers. Let $k$ be an algebraic number field, we denote by $\mathcal{O}_k$ the ring of algebraic integers in $k$. Let $K$ be a finite extension field of an algebraic number field $k$. Suppose for every ideal $I$ of $\mathcal{O}_k$, $I\mathcal{O}_K$ is principal. Then $K$ is called a PIT(Principal Ideal Theorem) field over $k$. Let $K$ be a PIT field over $k$. We say $K$ is a minimal PIT field over $k$ if $L/k$ is not PIT for every proper subextension $L/k$ of $K/k$.

(1) Let $k$ be an algebraic number field and $K/k$ be a finite extension. Is $K/k$ a minimal PIT if and only if $K/k$ is the Hilbert class field?

(2) Let $K/k$ and $L/k$ be minimal PITs. Are $K/k$ and $L/k$ isomorphic?