Considerations for frying a Turkey in Arctic Weather
I'm assuming you are using a turkey deep fryer and your question is what are the considerations in doing so in extreme cold. You can fry a turkey in extreme cold temperatures,  you just need to crank up the burner to make up for the heat loss. Your oil temperature is going to be 350&deg;F, if you were cooking it in +40&deg;F you'd be adding heat to raise the temperature 310&deg;F, in -40&deg;F you need to raise the temperature 390&deg;F instead, 390 is 125% of 310, so a rough approximation would be you need to add 25% more heat from the flame. It's the same principle in barbecuing, I've barbecued in -30&deg;F and you just need more heat, and to get the food inside asap when you're done! 

It would make sense to build in extra time if your equipment is going to be outside temperature as you will be handling things with gloves, and you have to heat up the pot more. If you can, keep the oil inside, or at least keep it above -10 so it doesn't gel. 

Also, you need to be sheltered from the wind when you deep fry outside in any temperature as wind will blow the burner heat away and disrupt the flame, it's even more important when it's that cold as wind will make the heat gap even higher, possibly to the point your burner can't make it up.