There's a thing called an easy-out. It's kind of a left-hand twist, tapered removal tool for just that type of thing. They come in different sizes according to the size screw, bolt, or stud you're trying to remove.

You drill a hole in the center of the broken screw with the appropriate size bit (one that's smaller than the diameter of the shaft the threads are on) and screw in the easy-out in counter clockwise. The tapered shaft has left-hand ridges on it that grab the inside of the hole you drilled causing it to dig in so you can remove the screw.

Pep Boys has them, and probably most good auto supply stores. Knowing the size screw that's broken would help you get the right size.

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Correct. I have a bunch of these in different sizes and I used them several times always with success. I got them at Sears (Craftsman brand) and it is called screw-out or screw remover.

i use the kit from home depot. its made by irwin and has about 6 different left-hand tapered extractors.

process is really simple. get a drill bit slightly smaller then the extracter. drill right through the broken screw so you have a nice hole for the extractor to dig into. put the extracter into that hole and give it a good tap with a hammer so its in, then get a pair of vice pliers and twist the extractor counter-clockwise. very simple procedure...