If k is a singular cardinal then there is no nonprincipal k-complete ideal over k.

Question

The problem stated in the subject is the exercise 7.7 of Thomas Jechs Set theory. Problem statement:
If $\omega_\alpha$ is singular, then there is no nonprincipal $\omega_\alpha$-complete ideal over $\omega_\alpha$.
I've thought about this problem for a few days now but i can't seem to solve it. Any kind of hint would do, anything to push me in the right direction. Thanks.
P.S. The problem used to have a hint... Atleast until my older brother crossed them out to annihilation with his pen. So if anybody has the hint of the book i would be grateful if you shared that too.

Answer 1

Sketch: Suppose that $\mathcal I$ is a nonprincipal $\kappa$-closed ideal on $\kappa$. First show that $$ \mathcal P_{< \kappa}(\kappa) := \{ X \subseteq \kappa \mid \mathrm{card}(X) < \kappa \} \subseteq \mathcal{I}. $$

Now observe the following: Since $\kappa$ is singular, there is some $\gamma < \kappa$ and a sequence $(X_i \mid i< \gamma)$ of sets $X_i \in \mathcal P_{< \kappa}(\kappa)$ such that $$ \kappa = \bigcup_{i < \gamma} X_i. $$ By the step above we have $X_i \in \mathcal I$. But, since $\gamma < \kappa$, this contradicts the fact that $\mathcal I$ is $\kappa$-closed.