The free group algebra appears often as a major tool in dealing with problems about free groups. It is also a lovely object for its own sake, featuring many analogies with the free group: for example, it satisfies an analog of the Nielsen-Schreier theorem. Yet, while many algorithms exist for different problems in free groups, there are very few algorithms for basic problems in the free group algebra.I plan to define the free group algebra, try to convince you it is a nice and interesting creature, and introduce a new algorithm for detecting whether a given element is "primitive" in a given ideal (what Whitehead's algorithm achieves in free groups).Based on joint work with Matan Seidel and Danielle Ernst-West.