[robpal]

I agree -- abstract algebra is elementary and some aspects of it can (and should), be taught in high-school. AA is a domain of maths which people won't even encounter if not studying a STEM field and such crash course could serve as a good outreach/vulgarisation material.

I really liked the materials in the link and think they are suitable for talented high-schoolers. Cox-Little-O'Shea is a fantastic book I studied as an undergrad and learned a lot from it, I wish someone would expose me to it earlier in life.

There's plenty of good math software for algebra which is not very popular (SageMath comes to mind as the most commonly used) and even more code that is simply inside knowledge. The problem is that people in the academia are not being paid for writing software but for publishing articles, even though the community value of a good package outweighs many papers. For example, good luck finding something that will compute non-commutative Groebner bases :)

[bkallus]

Macaulay2 will! https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1....

[madars]

Magma should do it (http://magma.maths.usyd.edu.au/magma/handbook/text/970 claims a Alan Steel's non-commutative generalization of Faugère F4 and noncommutative Buchberger algorithm) and US institutions can get it via the Simmons agreement. Not sure what's good for home users, though.

[lumost]

Curious why you find it so fundamental. In my experience abstract algebras primary benefit was drilling rigorous proofs for group theory etc.

Between physics and comp sci I'd rate abstract algebras contribution to understanding as one of the least important contributors ( although AA does provide a rigorous framework underpinning these fields in one way or another)

[jcranberry]

When I was in school, I felt it was very important because it taught me how to think of things in terms of mappings and structure-preserving mappings. After learning algebra found learning everything else became much easier.