Hmm, I can see ways of your ideia working, and based on all the information here you could build something very plausible with no magic involved.

I could see a civilization growing on a polar area, yet it would need some specific help of the enviroment. I would first include the rdanhenry's ideia, because indeed a underwater volcano area - we could even include geisers - would be like an oasis in the desert (and a polar region is basically a desert, except it is REALLY cold, instead of hot). This would make the area better for living. With this, you would have to think, there are not many plants that survive on a polar region, I mean, only some pines survive on north of Russia, for example, but your world is not earth, and your world doesn't have the same plants and animals Earth has - and TheHoarseWhisperer gave a good idea - so maybe it could have a plant capable of surviving on extremely cold regions, or even more than one, what would give the people there seeds/cereal/grains, or fruits, or at least wood. Of course the people would have to make a canals to the hot water get irrigate the plantations, and probably would need lots of care with the plants and stuff. The people could even have built a system to have plantations over the hot water - there are some ways, but you would need plants like rice which can grow with lots of water instead of dying like what happens with most plants.

Surviving in so cold regions would demand meat, of course, so they may have buffalos or Llamas, or another animal of your world that provides meat, wool, and maybe milk. Fish could be easly fished from the water next to the city. Fire can be come easily if you have pines, its sap (blood, I'm not sure what is right, used google translate) is extremely flamemable. With some mining - maybe an iron mine somewhere next, or faraway, from the city - they could have iron to make objects for both defense and plantation.

Remember, your world is not Earth, unless you want, so maybe it has no Tigers, but it has Ululos, which is a three headed cat 7 feet tall that eats oranges. Like what TheHoarseWhisperer said, you are looking for plausibility, not realism.