Polynomial Rings
Definition
Degree
Properties
Evaluation Homomorphism
This is a surjective ring homomorphism with if is commutative.
Factorization
Unique Factorization Domain (UFD)
Euclidean Domains
Examples
Example 1: is a Euclidean domain (with ).
Example 2: (for a field ) is a Euclidean domain (with ).
Example 3: (Gaussian integers) is a Euclidean domain.
Division Algorithm
Irreducibility Criteria
Gauss's Lemma
Applications
- Constructing field extensions
- Algebraic geometry (coordinate rings)
- Coding theory (error-correcting codes)
- Cryptography (finite fields)