I've been studying quite of analysis and while i was rewriting everything i was learning i didnt fully understand how to think of or how to visualize an "algebra", i could write it down and think about how to logically manipulate the symbols, but the true more abstract meaning came when i realized its interchangible with the word "field" which is a lot easier to say (1 rather than 3 syllables) and also has a natural intuitive interpretation as a physical field. So as far as I can tell its safe to say "field" rather than "algebra" most of the time.Have you guys any any interesting cases where you had to make a conscious choice between two usages of a word? Or did you just ignore it and use both?

with certain words in analysis, the choice of words is usually not a personal one, but rather one that's determined by the area of research. Analysts say "algebra" and "almost everywhere" and "convergence in measure" while probabilists say "field" and "almost surely" and "convergence in probability." When I first started learning probability theory, after having done a fair amount of analysis, I tended to use the analysts' terms (what's the point of using new terms just because they're used for spaces of measure 1?), but after a while, I just gave in. So although it doesn't matter that much, I figure you should choose your terms depending on whom you're communicating with.

"One must be able to say at all times -- instead of points, lines, and planes -- tables, chairs, and beer mugs." -- David Hilbert

In term of topology, an algebra is much more thant a field. You can look their precise definition in any maths book.The algebra has 3 laws whereas the field has 2, and has also a structur of vector space that a field has not.

QuoteOriginally posted by: JungixIn term of topology, an algebra is much more thant a field. You can look their precise definition in any maths book.The algebra has 3 laws whereas the field has 2, and has also a structur of vector space that a field has not.Well, again, it depends on what area of math you're referring to. If you look in a textbook on modern algebra, you'll find definitions for an "algebra" (which is essentially a ring with a vector space structure) and a "field" (essentially a ring with additive and multiplicative identities and inverses) which are definitely different from each other, and also these are completely different concepts from the definitions you'll find in an analysis or probability textbook for an "algebra" or a "field" (a collection of subsets of a set which is closed under taking complements and unions). Durrett's graduate probability textbook, for example, defines an algebra and a field to mean the same thing. Also, the analysis/probability definition of an algebra/field has only two laws (closure under complements, and under unions); the third law that one usually sees---closure under countable unions---requires the qualifier "sigma" in front of "algebra/field."