Algebraic Integers of a Field

Question

Suppose my ring is $\mathbb{Z}[e^{i2\pi/p^2}]$. I wish to prove that the algebraic integers are ${\sum_{n = 0}^{p^2-p-1}a_{n}e^{i2n\pi/p^2}}$ where each $a_{k} \in \mathbb{Z}$. I see why thus finite sums are algebraic integers: every integer is algebraic integer and the complex exponent is a root of $1+x^p+x^{2p} + ... + x^{p(p-1)}$ so it is algebraic so the the sum is algebraic integer because the algebraic integers form a ring. How do I show that every algebraic integer in my ring has this form? That is why must each $a_{n}$ be an integer?