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We will study the core objects in abstract algebra: groups, rings and fields. The term ``abstract'' highlights the approach to mathematics that emerged in the 20th century in which there was less reference to ``real objects'' and a greater emphasis on proving theorems from a small set of simple axioms. For example, the symmetries of a 3-dimensional object tell you something about the nature of the object. But, symmetry itself is interesting. Symmetries of an object can be composed (apply one then another) and produce another symmetry. Each symmetry has an inverse, which undoes the given symmetry. Abstractly, one can consider the properties of a set (e.g. symmetries) with an operation (e.g. composition) and an inverse to the operation. We call such an object a group. Discovering the structure and properties of abstracts groups was a massive project in the last century.
Ring theory evolves from recognizing that the basic properties of the integers (from grade school!)---two operations ($+, *$), with commutativity, associativity, distributivity---extend to polynomials and can be extended further to abstractly defined objects. Similarly, the properties of the rational numbers (now every nonzero element has a multiplicative inverse) can be considered in an abstract axiomatic way. This is the area called field theory.
I love this material because it gives a deeper and richer understanding of school mathematics, but also unveils a world of very complex and intricate beauty.
Required.
(Recommended) This has been the standard text for undergraduate algebra at SDSU for a few years.
(Recommended) Available online.
(Alternative possibility) A widely used and well written text that can also be a reference.
(Alternative possibility) This is another good undergraduate algebra text. Available free online. See the button for "Sage and AATA." Download "Sage Worksheet Collection." Open the zip file from the sage notebook.
Some of the following topics will be familiar from your undergraduate course. We will cover them in greater depth, and with more attention to details. I expect this material to take 9 weeks.