**Ring Theory Notes PDF**

*Date: 27th Jan 2023*

In these “** Ring Theory Notes pdf**”, we will study the basic concepts of ring of polynomials and irreducibility tests for polynomials over ring of integers, used in finite fields with applications in cryptography. This course emphasizes the application of techniques using the adjoint of a linear operator and their properties to least-squares approximation and minimal solutions to systems of linear equations.

We have provided multiple complete ring theory notes for bsc for any university student of BCA, MCA, B.Sc, B.Tech CSE, M.Tech branch to enhance more knowledge about the subject and to score better marks in the exam. Students can easily make use of all these ring theory notes for bsc by downloading them.

**Topics in our ring theory handwritten notes pdf**

The topics we will cover in these **ring theory handwritten notes pdf** will be taken from the following list:

**Polynomial Rings and Unique Factorization Domain (UFD):** Polynomial rings over commutative rings, Division algorithm and consequences, Principal ideal domains, Factorization of polynomials, Reducibility tests, Irreducibility tests, Eisenstein’s criterion, Unique factorization in Z[x]; Divisibility in integral domains, Irreducibles, Primes, Unique factorization domains, Euclidean domains.

**Dual Spaces and Diagonalizable Operators:** Dual spaces, Double dual, Dual basis, Transpose of a linear transformation and its matrix in the dual basis, Annihilators; Eigenvalues, Eigenvectors, Eigenspaces and characteristic polynomial of a linear operator; Diagonalizability, Invariant subspaces, and Cayley-Hamilton theorem; Minimal polynomial for a linear operator.

**Inner Product Spaces:** Inner product spaces and norms, Orthonormal basis, Gram-Schmidt orthogonalization process, Orthogonal complements, Bessel’s inequality.

**Adjoint Operators and Their Properties:** Adjoint of a linear operator, Least-squares approximation, Minimal solutions to systems of linear equations, Normal, self-adjoint, unitary, and orthogonal operators and their properties.

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