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Once I have set up my IDT with all the interrupts, have loaded the IDT, entered pmode, reset the PIC, and set up an interrupt handler, will I be able to access interrupts in protected mode?

Or will I have to switch to real mode, unreal mode, or v86 mode to still use interrupts?

If you are in PMode and have a valid IDT with interrupt handlers, you can of course make a Software Interrupt and receive Hardware Interrupts...

you can handle exceptions and interrupts. however, you can not call bios interrupts directly without being in v86 mode or rmode