No, you don't need to buy Flash. The Flex SDK has a compiler. You still might want to get Flex Builder (or the upcoming Flash Builder, which is the same as Flex Builder, but very different than Flash--not confusing at all...). Or I think there are some third party IDEs for Flex, one or more of which might be free, but I'm not sure. I imagine it would be pretty hard for someone just learning flex and air to start building apps without Flex Builder. Also, I think the beta of Flash Builder is still available for free, if you want to try it out.