Often when I'm working on a math problem, there comes a point when I've set everything up and what r

Question

Often when I'm working on a math problem, there comes a point when I've set everything up and what remains is to expand some expression, substitute something in, solve an equation, or otherwise enter the domain of algebra.
I find that I usually think about this step in the problem as a black box, into which I put a set of equations and out of which comes a set of solutions. This is particularly true if the algebra involved includes many steps.
This is good in a lot of ways and bad in some others, but putting that aside, I'm fascinated that this step seems to come up in all areas of math. Is it possible to separate math from algebra? Are there any branches of math where long chains of algebra are not found or uncommon?

Answer 1

To be honest, I think it depends what you mean by Algebra. In my experiences with some of my classes, the professors stop caring about complicated arithmetic. If there comes some point of a proof where complicated arithmetic, or the solution of a complicated system of equations is needed, we are allowed/encouraged to use computer algebra software.
In general, I think that most tedious arithmetic can be avoided, at least in my upper level undergraduate courses- I can't necessarily speak for the guys doing graduate coursework or beyond.
My experience of Abstract Algebra and Galois Theory was devoid of arithmetic, although the theory behind it can help understand or simplify arithmetic computations. Most of my upper level classes have been, as well.

Often when I'm working on a math problem, there comes a point when I've set everything up and what r

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