# Why is the exterior algebra called the “exterior algebra?” What makes it “exterior?”

Why is the exterior algebra called the "exterior algebra?" What makes it "exterior?" Is it just because a module can be universally embedded into its exterior algebra, so one could view the exterior algebra as surrounding the module? Why is it not just called the "alternating algebra?"

• Probably hard to track down, especially since the reason might be a foreign-language thing. Grassmann referred to aussere produkt and you can read about äußeren Algebra in German wikipedia. I don't speak German, but it seems like from the start "external/outer/exterior" was used. English wikipedia offers this (speculative, IMO) reason: The term "exterior" comes from the exterior product of two vectors not being a vector. In any case, this is not a math question, it is one for historians. – rschwieb Sep 8 '17 at 15:25
• @rschwieb well, we do have the math-history tag (which explicitly targets evolution of terminology). – Thomas Sep 8 '17 at 15:33
• @rschwieb fine with me. This one is still there, though. Edit: and, actually, who is the 'we' you are referring to? This site is run by its participants. So as far as I'm concerned, I'm part of 'we', As is the OP. So whether 'we' will phase out or not -- we'll see.... – Thomas Sep 8 '17 at 18:23
• Dear @Thomas : Perhaps more to the point, maybe I should just say that the mere existence of a tag does not justify questions using the tag for all eternity, which is why I mentioned we sometimes phase out tags. Regards – rschwieb Sep 8 '17 at 18:36
• @Thomas the phasing out does not necessarily result in removal of the tag. What's more the tag description does in fact already recommend the other site. Moreover, there are also mathematical questions that have a historically aspect to them for which the tag is useful to keep around. Note that the site also has a physics tag, while physics proper is certainly not on-topic. – quid Sep 8 '17 at 21:12

It was Grassmann that called it exterior because to have a non-null product the multiplicands must be geometrically one to the exterior of the other. For instance $$\mathbf{x}\wedge\mathbf{y}\wedge\mathbf{z}=0$$ if $\mathbf{x}$ lies in (is not exterior of) the subspace spanned by the $\mathbf{y}$ and $\mathbf{z}$. So the product is called exterior product, and consequently the algebra with this product is called exterior algebra.