- Set Theory and Arithmetic
Toward the end of the nineteenth century, a branch of mathematics called "set theory" was developed by the German mathematician George Cantor. Although Cantor's original investigations were undertaken in order to clarify the notion of infinity in mathematics, it was soon discovered that the theory of sets provided a convenient framework in which to discuss all of mathematics. More precisely, it was discovered that one can deduce all known mathematics from a list of axioms about sets. Thus it is not surprising to find that set theory has become an indispensable tool in discussing all of mathematics, algebra included. In the following sections we will describe some of the most basic facts about set theory and their application to the study of the most fundamental algebraic system-the integers. However, we will not embark on a formal study of set theory and its relations to the foundations of mathematics. Rather, for us, set theory will be merely a convenient language in which to state algebraic results.
- The Theory of Groups
In this chapter we begin the study of one of the most beautiful branches of mathematics - the theory of groups. A group is one of the most elementary mathematical systems, having only one operation defined among its elements. Therefore, from a logical point of view, groups should be one of the first algebraic systems to be studied. Such study, however, would be wasted if groups did not occur "in nature." We have already hinted at the connection between groups and the solution of equations in radicals. We will see that groups occur with surprising frequency in mathematics, and this is precisely the reason that the notion of a group is one of the central notions of modern mathematics.
Our approach to group theory is "abstract", that is, we will define a group by a set of axioms and then deduce general properties of groups from the axioms. When the general theorems are applied to particular groups, we are able to determine data about individual groups. The reader should recognize that this approach is the reverse of usual historical discovery. Mathematicians usually discover theorems by generalizing particular examples or known special cases. The important point for the beginner to recognize is that mathematicians do not just write down random systems of axioms and then proceed to write theorems based on these axioms. Not all axiom systems are "interesting." In order for an axiom system to be interesting there must be important examples to which the axioms apply.
- The Theory of Rings
In this section we will pursue a similar plan to that carried out in the previous sections and will develop the theory of the algebraic system called a ring. In the section on sets we devoted considerable energy studying the properties of the integers Z. We subsequently found that with respect to the operation of addition, Z is an abelian group. However, Z is much more. In Z, there are two operations; + and ·. Each of these operations is associative and commutative and has an identity. Each element a of Z has an inverse -a with respect to addition. Finally, the two operations are connected via the distributive laws:
a·(b + c) = a · b + a · c,
(b + c) · a = b · a + c · a (a,b,c Z
In this section we will abstract from the properties of Z the notion of a ring and then study the properties of rings. Many of the phenomena which we have already observed in Z will carry over to other rings. And this is precisely the power of the 'abstract approach' to the theory of rings.
- Unique Factorization
- Vector Spaces
In the preceding chapters we have studied various algebraic structures, including groups, rings, and fields. We will now introduce another algebraic structure, which will play an important role in our development of the the theory of equations. This algebraic structure - the vector space - is motivated by the consideration of vectors in physics. However, its importance is felt in many branches of and in many disciplines to which mathematics is applied.
- The Theory of Fields
In this chapter we will begin to study the theory of fields, with an eye toward the Galois theory of equations. As a consequence of the theory which we will develop in this chapter, we will be able to attack the classical geometrical problems concerning constructions with compass and straightedge. In particular, we will show it is impossible to trisect the general angle or duplicate the cube, using only compass and straightedge.
- Galois Theory
We shall now examine more closely the modern theory of equations. This is the Galois theory of fields. It is a beautiful abstract interplay of fields and groups and helps to solve many interesting problems, such as why there are no analytic solutions to polynomials of degree 5 and above.
In this section you will find items of interest that do not flow with the rest of the text or are articles, while outside the scope of the text, may be of interest to the reader