Second lecture continues with the Discrete Logarithm Problem and how it establishes the foundation for the Diffie-Hellman Key Exchange The main focus is the structure of finite fields, field characteristics, formal polynomial operations (long division), and the construction of Galois fields, $GF(p^n)$
As an exercise, try verifying that the addition and multiplication tables for GF(4) actually satisfy the mathematical axioms of a field, calculating the multiplicative inverses.
You can also practice taking a polynomial over F2, dividing it by the prime polynomial $x^2 + x + 1$, and finding the remainder to see exactly how the elements of GF(4) are constructed manually.
Multiplication tables for GF(4) construction step-by-step: math.stackexchange.com/questions/3660825/how-are-the-addition-and-multiplication-tables-for-gf4-constructed
Review the corresponding chapters in the course book here: Google Drive Link