Why doesn't it rain in the Arctic Circle?

Answer 1

The question is kind of backward.

The reason it's a desert is because it doesn't rain much there.

The reason it doesn't rain much in some places (which we call deserts) has to do with wind patterns and geographical features and is fairly complicated.

Answer 2

Within the arctic circle it is generally too cold for rain: any precipitation wthat falls will be frozen. Most of it will be snow.