Polynomials

Explore the fundamental concepts of polynomials in this comprehensive lecture. Delve into formal definitions of fields and finite fields before examining polynomials in depth. Learn about adding, multiplying, and dividing polynomials, as well as the concept of "division with remainder." Investigate polynomials in F[x], polynomial evaluation, and roots. Discover Lagrange Interpolation and various methods of representing polynomials. Gain insights into practical applications, including sending messages on noisy channels and using repetition codes. This in-depth exploration provides a solid foundation for understanding polynomials and their diverse applications in mathematics and communication theory.

Intro
Field - formal definition
Finite fields
Polynomials - formal definition
Adding and multiplying polynomials
Dividing polynomials?
"Division with remainder" for polynomials
Polynomials F[x]
Evaluating polynomials
Polynomial roots
Lagrange Interpolation
Recall: Interpolation
Representing Polynomials
Sending messages on a noisy channel
Simpler case: "Erasures"
Repetition code - noisy channel
Enter polynomials