this is a vague question but what’s a good way to develop algebraic intuition?
i have pretty good analysis intuition (dipped my toes into most grad analysis topics – measure theory, spectral theory, pdes, optimization, sdes, etc.) but for the life of me even undergrad algebra things are sometimes unintuitive to me (often when they intersect with number theory type things).
obviously i’m aware of the age old adage of just doing problems and i’ve tried – i have i think 4? algebra books ranging in difficulty from frayleigh to ashe – but i just can’t make substantial progress because i’m constantly stymied by seemingly trivial steps in proofs.
a little while ago i had the idea that maybe playing with the objects using something like magma would be useful but i can’t afford a magma subscription just for fun.
also i thought about approaching it from the other direction – cat theory through something practical like haskell. but the trouble there is (as everyone says) cat theory doesn’t make sense unless you have algebraic examples to translate to.
i’ll give an example that i should probably be embarrassed about but since i have no mathematical shame i’m not: i’ve read the definitions of group, ring, field, algebra, probably over 100 times each. i still for the life of me can never remember which ones have units and/or inverses for each of the operations (though i do know which ones have two operations) and which ones don’t. like there’s nothing about the word ring (i just now had to look it up) to me that leaps out as addition should be invertible but multiplication need not be.
it’s like i’m one of those people that immediately forgets your name after you introduce yourself to me (which is in fact true for me) but for algebraic definitions. note i do not have this problem in the least in analysis because i always have a picture of what’s going on. e.g. my last analysis class was ~10 years ago and i can define uniform and lipshitz continuity without skipping a beat (though i don’t remember absolute continuity i realize).
now i’ve gotten away with this because i’m not a math phd (CS) but still there are things i’m personally interested in (like de rham cohomology) that just demand facility with algebra.
anyway this isn’t a rant or a cry for help – i’m genuinely interested in ideas for other ways to approach this seemingly ludicrous learning … “disability”.