  In this paper, we study the partition theory over totally real number fields.
Let $K$ be a totally real number field. A partition of a totally positive
algebraic integer $\delta$ over $K$ is
$\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ for some totally positive
integers $\lambda_i$ such that $\delta=\lambda_1+\lambda_2+\cdots+\lambda_r$.
We find an identity to explain the number of partitions of $\delta$ whose parts
do not belong to a given ideal $\mathfrak a$. We obtain a generalization of the
Euler-Glaisher Theorem over totally real number fields as a corollary. We also
prove that the number of solutions to the equation
$\delta=x_1+2x_2+\cdots+nx_n$ with $x_i$ totally positive or $0$ is equal to
that of chain partitions of $\delta$. A chain partition of $\delta$ is a
partition $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ of $\delta$ such
that $\lambda_{i+1}-\lambda_i$ is totally positive or $0$.
