You are already familiar with the real numbers and the use of these laws in Secondary and College Algebra. But a math major or a math teacher needs to have a deeper knowledge than just College Algebra. There are two strategies we will follow in developing a deeper knowledge of these rules. The first is to look at other examples of fields, that is other algebraic structures that satisfy the nine rules listed above. We will look at the field of rational numbers, the field of complex numbers, and several examples of finite fields. A second way to study the field laws is to see what happens if we drop some of them. In addition to studying fields, we will study domains, like the integers under addition and multiplication, rings, like the set of 2´ 2 matrices under matrix addition and matrix multiplication, and groups, like the symmetries of a square under composition. Each of these algebraic structures satisfies a partial set of the field laws. Comparing different fields to the familiar field of real numbers and seeing what happens when one of the field laws is dropped will give you a deeper appreciation for what the different laws mean.
There are two classes at the 500 level in the mathematics department that cover this material. One is this class, Math 511, and the other is Math 512, Introduction to Modern Algebra. Math 512 will focus on theoretical matters. Math majors are recommended to take Math 512, especially if there is any chance of your pursuing graduate studies in mathematics. On the counseling exam for this year’s entering graduate class, students were asked to prove a simple fact about products of elements of normal subgroups with trivial intersection. Students who couldn’t answer that question got placed in Math 512. At the end of this class, that question won’t make any more sense to you then it does now. If you ever head to graduate school, you will need to take Math 512, so you might as well take it now.
This class, Math 511, is designed for students in Mathematics Education. Our focus will be on applications (though we won't ignore theory). We will start with a real-world problem (or at least what a mathematician thinks of as a real-world problem) and work through the solution. Along the way, we will need to develop properties of various algebraic structures. At the end, you should have a feeling for the roles of all of the field laws that you will be teaching in secondary school algebra and also an answer for students who want to know what all this is used for. Applications I’m planning on studying this semester include: How can you transmit information securely over the World Wide Web? How does a CD player store information so the music can keep playing even if a scratch obscures part of the disk? How can you prove that it is impossible to trisect an angle with straightedge and compass? These questions come with several variations and we will probably cover some additional topics as well.
Grading in this class will be based on homework, in-class exams, and a final exam. There will probably be two in-class exams at dates to be announced later. The final exam will be Thursday, December 17, from 11:50am to 1:40pm. Your homework average, the total of your scores on the two exams, and your score on the final will be equally weighted in the grading.
Plagiarism and cheating are serious offenses and may be punished by failure on the exam, paper or project, failure in the course, and/or expulsion from the University. For more information refer to Appendix F in the Faculty Handbook.
If you have any condition, such as a physical or learning disability, which will make it difficult for you to carry out the work as I have outlined it or which will require academic accommodations, please notify me in the first two weeks of the course.