Normal $\kappa$-complete non-principal ultrafilter on measurable cardinal $\kappa$

Question

Is it true that for every measurable cardinal $\kappa$ there is a normal, $\kappa$-complete, and non-principal ultrafilter on $\kappa$ ?

Answer 1

Yes. This can be shown in a more purely combinatorial way, but it's probably best to follow the proofs of

1. If $\kappa$ is measurable, then $\kappa$ is the critical point (i.e. least ordinal moved) of some nontrivial elementary embedding $j:V\to M$ where $M$ is an inner model.
2. If $j:V\to M$ is a nontrivial elementary embedding with critical point $\kappa$, then $\{X\subseteq \kappa: \kappa\in j(X)\}$ is a normal ultrafilter on $\kappa.$

This is all in Jech (as is a proof not using the elementary embedding characterization).