Algebra

Algebra is perhaps best defined as the study of “algebraic structures.” Broadly construed, algebraic structures are sets endowed with additional operations—think of the integers (a set) with addition and multiplication (additional operations). Usually, these operations allow elements to be combined in some way, but sometimes they’re better described as allowing transformation of elements into each other—as, for instance, in the case of the category of sets endowed with the usual mappings between sets. Such structures are ubiquitous: integers and their various extensions, the vector spaces you studied in linear algebra, groups of transformations that encode symmetries, abstract categories that capture relationships between complex objects (like groups themselves) are all examples. Due to this ubiquity, algebra crops up in all areas of mathematics and has a host of applications.

The first two structures you are likely to encounter are groups and rings. In general, groups should be thought of as collections of transformations. That’s because objects in a group can be composed (usually called “multiplication”) and inverted, just like functions, but nothing else is required for a set to have the structure of a group. Groups are important and beautiful objects. For instance, the set of all invertible linear transformations of a vector space to itself is a group, as is the set of all rotations of space (of any number of dimensions). These examples also illustrate the interpretation of groups as symmetries of some other mathematical entity—in this case, a vector space. Groups also play this role in the setting of permutations (symmetries of finite sets), geometric rearrangements of shapes (symmetries of polygons), and Galois theory (symmetries of number systems).

This last example shows how groups can be connected to their more sophisticated cousins, rings. While a ring can be thought of as a group with additional structure, a ring is really a very different animal. Rather than functions and transformations, rings represent numbers. In this case, you can add them, take their negatives, and multiply them—but that’s it. Since many natural number systems lack division (is one divided by two an integer?), it is excluded from the defining properties of rings. When a ring has it anyway, it is called a field—thus, we speak of the “ring of integers,” but the “field of rational (or real, or complex) numbers.” Rings serve as a natural generalization of the integers, and they provide a setting for the study both of abstract objects that resemble them and concrete number systems extending them, like the rational numbers, or the set of all numbers of the form a + b √5 with a, b integers.

Now, all of this is just the beginning. We haven’t even mentioned the question of commutativity, or the fact that there is a natural way to think of rings as collections of functions (just ask an algebraic geometer). Nor have we really discussed the rich theory that results in studying the way algebraic objects interact with other mathematical entities. Nor the details of Galois theory. Nor categories. The list goes on and on. But Princeton offers many ways for you to get into these and other algebraic topics. This begins with an introductory sequence that starts with groups, develops the fundamental aspects of the theory behind them, then builds up elementary ring theory and the most important parts of classical Galois theory.

After you’ve completed the introductory sequence, you will be adequately prepared to jump into the more advanced courses that investigate the topics we’ve mentioned above.

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