In Kummer's theorem, it is possible to write ideal factorization as- $$\langle x + y\rangle\langle x + y\zeta\rangle···\langle x + y\zeta^{p−1}\rangle = \langle z^p\rangle \cdots (1)$$ in which all factors are interpreted as principal ideals, here, $\zeta = e^{\frac{2πi}{p}}$, $\langle a \rangle$ is an ideal. which is dreied from $$(x + y)(x + y\zeta)···(x + y\zeta^{p−1}) = z^p \cdots (2)$$
But if $\langle a \rangle$ is an ideal, it represents all multiple of $a$ which makes the ideal $\langle a \rangle$ an infinite set, then how can we write an equation (1) from equation (2)?
Doesn't the left hand and right hand side of equation (1) represent infinite set?
What am I missing? Please explain.