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Introduction to Modern Algebra
When you learned arithmetic, you learned to calculate using numbers and the four operations - addition
- subtraction
- multiplication
- division.
For example, you learned that
- 2+2=4
- 0*7 = 0
- (-1)*(-1) = 1
- You cannot divide by zero!
But did you ever ask
- Why?
- Does it have to be this way?
- Could there be a different answer?
- Can we divide by zero?
- Do two negatives always make a positive?
In Modern Algebra, we will answer these questions and many more by looking at the structure of mathematical systems. By looking at many different examples of mathematical structures, we will begin to see things that are true about every structure. This is why we sometimes call this course "abstract algebra."
Each mathematical structure that we will study is a set with one or more operations. The basic structures at which we will look are
- Groups: Sets like integers with addition/subtraction.
- Rings: Sets like integers with addition/subtraction and multiplication (but not division).
- Fields: Sets like rational numbers with addition/subtraction and multiplication/division (except 0)
Some of the interesting specific questions we will address are
- How do credit card machines know if you entered a legitimate card number?
- How many ways are there rotate a soccer ball so that no one can tell that it has been moved?
- How can you send a message that no one (not even the sender) can decode except the receiver?
- Why is the product of two negative numbers always positive?
- Why can't you trisect any angle with a straight edge and compass?
In the process, you will learn to
- Calculate using a variety of mathematical objects and operations (not just numbers or polynomials).
- Generalize and prove conjectures.
- Make a giant leap toward being a mature mathematician.