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As far as I can tell, I've set up a GDT, and it seems that the logical thing to do now is to set up an IDT.
How can I check if my GDT does in fact work? My kernel does run, without any obvious errors.
I need to know if it works before implementing an IDT, right? Do I just have to trust that it does?

You can check the segment limits etc. in bochs or some other debugger. But other than that, your GDT works if your kernel does not raise random general protection exceptions or triple fault.