Then a solid state relay would work fine and would be less than $10. One side takes a DC input (you'd connect the + side to an Arduino digital pin and the other side to the Arduino GND) and the AC goes on the other side. Make sure it's rated greater than 15A AC of course. "Fotek" is a popular (cheap) brand that you can find on Amazon or eBay. If you google for "Sous Vide" cookers you'll assuredly find lots of examples of them in use.
You can also use a typical electromechanical relay, but SSRs are getting so inexpensive it's not worth the money savings if you're just switching a single AC line. SSRs have no moving parts and are preferred if you're doing a lot of switching on/off as you might be doing if you want to keep the temperature in a very narrow band.