Why do books titled “Abstract Algebra” mostly deal with groups/rings/fields?

As a computer science graduate who had only a basic course in abstract algebra, I want to study some abstract algebra in my free time. I’ve been looking through some books on the topic, and most seem to ‘only’ cover groups, rings and fields. Why is this the case? It seems to me you’d want to study simpler structures like semigroups, too. Especially looking at Wikipedia, there seems to be a huge zoo of different kinds of semigroups.

Answer

There is not that much substantial to say about general semigroups at an introductory level, in comparison to groups, say, and what there is to say at this level (e.g. the theory of Green’s relations) tends to rely on a prior understanding of groups in any case. (One thing I have in mind here — but I am speaking as a non-expert — is that semi-groups are much closer to general universal algebra than groups, and universal algebra is not, I would say, appropriate as a topic for a first course in abstract algebra.)

The theory of groups lends itself well to a first course, because the axioms are fairly simple but lead fairly quickly to non-trivial theorems, such as the Sylow theorems. Groups (especially group actions) are also ubiquitous, and the kind of counting arguments which one can make are very useful to learn. (It is the inability to make these sorts of counting arguments that causes so many problems when you investigate semi-groups, in comparison to the case of groups.)

The theory of fields is particularly useful in number theory, and Galois theory also provides a nice tie-in with the theory of groups, which in fact served as Galois’s motivation for introducing groups in the first place. The theory of fields (especially the part to do with finding roots of polynomials) is also the part of abstract algebra that is closest to high school algebra, so it is not surprising that there is a substantial focus on it.

The theory of rings normally appears both because it is a precursor to field theory (fields are particular kinds of rings, and the polynomial rings $F[x]$ also play an important role in the study of fields), and because it includes many basic examples from mathematics, such as matrix rings, the integers, quaternions, and so on. Rings also play an important role in the study of group representations (via the appearance of group rings), which, even if they don’t appear in a first course, are just over the horizon.

Summary:
So overall, I think the answer is that groups, rings, and fields are the parts of algebra that are most closely connected to the basic core topics of mathematics, and are also closely integrated with one another. (Groups not
immediately obviously so, but because of Galois theory and group rings,
for example.) The theory of semigroups, by contrast, doesn’t play much role in the rest of mathematics, and the theory that does exist is more complicated than the theory of groups (despite the axioms being simpler).

Attribution
Source : Link , Question Author : GeorgW , Answer Author : Matt E

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