It is unexceptional (indeed, expected) to get through an American undergraduate science or engineering degree without ever taking an abstract algebra course (much less the 2+ apparently expected of German pure math students).
But in any event, the top post here by Garlef is barking up the wrong tree. Division algebras, field extensions, and galois theory (per se) are not the tools to use for studying arbitrary-dimensional geometry. What you want is Clifford algebra (which Clifford himself, and later Hestenes, call “geometric algebra”) and then geometric calculus, which can be used on arbitrary manifolds, in non-metrical contexts, etc.
Basic geometric algebra should be taught to advanced high school students and all undergraduates studying any technical subject.
Math students looking for a math-style introduction to geometric algebra should try Chisolm (2012) https://arxiv.org/abs/1205.5935
