Mathematics Foundations/7.4 Polynomial Rings

7.4 Polynomial Rings 7.4.1 Introduction to Polynomial Rings A polynomial ring is a ring formed from the set of all polynomials in one or more variables with coefficients in another ring, typically a field. If is a ring, the ring of polynomials in with coefficients in is denoted by . Definition A polynomial over a ring is an expression of the form: where and is a non-negative integer. The elements are called the coefficients of the polynomial. The largest value of for which is called the degree of the polynomial, denoted by . 7.4.2 Algebraic Structure Ring Properties The set forms a ring with the following operations: Addition: Multiplication: where and . Important Properties If is commutative, then is commutative. If is an integral domain, then is an integral domain. If is a field, then is a principal ideal domain (PID). For polynomials and with coefficients in an integral domain, . 7.4.3 Division Algorithm When the coefficient ring is a field, we have a division algorithm analogous to the one for integers. Theorem (Division Algorithm) Let be a field and let with . Then there exist unique polynomials such that: where either or . This leads to concepts of divisibility in similar to those in the integers. 7.4.4 Irreducible Polynomials Definition A non-constant polynomial is irreducible over if it cannot be expressed as a product of two non-constant polynomials in . Irreducible polynomials play the role of "prime elements" in polynomial rings. Examples is irreducible over but reducible over as . is irreducible over but reducible over as . 7.4.5 Unique Factorization Theorem If is a field, then every non-zero polynomial can be written uniquely (up to units and the order of factors) as a product of irreducible polynomials: where is a non-zero constant, are monic irreducible polynomials, and are positive integers. This makes a unique factorization domain (UFD). 7.4.6 Polynomial Rings in Multiple Variables The construction of polynomial rings can be extended to multiple variables, denoted as . Definition A polynomial in variables over a ring is an expression of the form: where all but finitely many of the coefficients are zero. Properties can be viewed as . If is a field, is not a PID for , but it remains a UFD. 7.4.7 Applications and Connections Polynomial rings have numerous applications and connections to other areas of mathematics: Field extensions and Galois theory Algebraic geometry (polynomial rings as coordinate rings) Coding theory (error-correcting codes) Cryptography (polynomial-based cryptosystems) Solving algebraic equations 7.4.8 Quotient Rings and Field Extensions If is a field and is an irreducible polynomial in , then the quotient ring is a field that contains as a subfield. Example The field can be constructed as , where we identify the equivalence class of with the imaginary unit . 7.4.9 Exercises Prove that if is an integral domain, then is an integral domain. Find all irreducible polynomials of degree at most 3 in . Determine whether is irreducible over . If and , find polynomials and such that with . Prove that if is a field of characteristic 0, then any polynomial of the form where is prime is either irreducible or has a root in .