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For my Abstract Algebra class, we will be doing small presentations (2 class periods) covering some topic in Abstract Algebra. Thus far, I have studied groups, rings, fields, modules, tensor products, exact sequences, algebras, some basic category theory, and some other things.

This semester we are presenting some subtopic of Abstract Algebra (presumably an extension of what we have already learned).

What are some topics you would suggest and why?

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This should be community wiki ... –  Michael Joyce Apr 9 at 17:24

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Another topic could be the study of the three famous geometric constructions with compass and straight edge, or better, that they cannot be constructed as such: Squaring a circle, doubling the volume of a cube and trisecting an angle. For that you need rings and fields and all sorts of abstract algebra stuff. Did it a long time ago, forgotten for the most part, but this lingered in my mind.

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Geometric constructions are a cool application of field theory. For example, if a number $\alpha$ is constructible as a length, then the field extension $\mathbb{Q}[\alpha]$ is a degree power-of-two field extension over the rationals.

Further, asking whether a regular $n$-gon is constructible with a compass and straightedge is equivalent to asking whether the $n$th cyclotomic field can be built as a tower of quadratic field extensions.

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The impact of Algebra on Number Theory (and vice-versa!) is considerable.

Several theorems in Number Theory have nice algebraic proofs.

For instance, Fermat's theorem on which numbers can be expressed as sums of two squares has a nice proof in terms of factorisation in the PID $\mathbb Z[i]$.

See also Connections between number theory and abstract algebra.

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One topic that I find interesting is finite reflection groups. They have a lot of importance in studying highly symmetric geometric spaces.

A related, but distinct, topic would be to discuss the 17 wallpaper groups and illustrate the various patterns that are possible.

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I find semigroup theory interesting: it's rich and have applications to computer science (mainly in automata theory). See for instance's Green relations.

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