Suppose an object (or a concept or ...) is named after the person
X, in honor of Mr. or Mrs. X in mathematics:
It is natural for me to write the first letter of his/ her name in English in capital letters when referring to that object/ concept. For instance: Gaussian curvature, Newtonian mechanics, Riemannian geometry, Archimedean valuation, Eulerian graph, Platonic solids, Jacobian matrix, Noetherian rings, Artinian rings, Hamiltonian path, Hermitian matrix, Hessian matrix, ...
But why do we usually write
abeliangroup instead of
Abeliangroup? Or why do we usually write
abelianvariety instead of
My strong suspicion is that perhaps the importance of abelian groups has reached us through mathematicians in languages other than English. But since I do not have a background in mathematical history, it is very likely that my guess is wrong.
Is this related to the French school of mathematics? (I know that "variety" is the French equivalent of "Manifold".) If yes, then why do we write Galois extensions?
Are there any exceptions other than abelian groups?
Also, I do not know what is the suitable tag for my question.
I thought that perhaps the word abelian had entered the "ordinary human language", and had been accepted as an adjective. (Why did I use the term "accepted"? Because without losing accuracy, we can easily translate this word into "ordinary human language". In some sense, it has replaced the word commutative in some areas of the mathematical literature.)
That's why I checked ablian's definition from Merriam-Webster's dictionary, and this hypothesis was reinforced for me. (It states that: "adjective, often capitalized") Also, the status of the word euclidean is the same as the status of the word abelian. (It is mentioned again that: adjective, often capitalized) Also, see platonic.
When I came here to add my observations to my questions, I saw Professor K. Conrad's comment, in which he introduced an MO post (Why is “abelian” infrequently capitalized?), which is exactly the subject. I saw the answers there, and the discussions there were very convincing to me.