Do groups, rings and fields have practical applications in CS? If so, what are some?

This is ONE thing about my undergraduate studies in computer science that I haven’t been able to ‘link’ in my real life (academic and professional). Almost everything I studied I’ve observed be applied (directly or indirectly) or has given me Aha! moments understanding the principles behind the applications.

Groups, Rings and Fields have always eluded me. I always thought they were useful (instinctively) but failed to see where/how. Are they just theoretical concepts without practical applications? I hope not. So what are their applications, especially in the field of computer science. No matter how arcane/remote their use I still want to know.

Answer

If every second or so your computer’s memory were wiped completely
clean, except for the input data; the clock; a static, unchanging
program; and a counter that could only be set to 1, 2, 3, 4, or 5, it
would still be possible (given enough time) to carry out an
arbitrarily long computation — just as if the memory weren’t being
wiped clean each second. This is almost certainly not true if the
counter could only be set to 1, 2, 3, or 4. The reason 5 is special
here is pretty much the same reason it’s special in Galois’ proof of
the unsolvability of the quintic equation.

-Scott Aaronson

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dessin d’enfant terrible

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