Mathematics Foundations/7.2 Rings and Fields

Rings and Fields 7.2 Rings and Fields Introduction After studying groups in the previous section, we now turn our attention to two other fundamental algebraic structures: rings and fields. These structures build upon the concept of groups by introducing additional operations and properties, allowing us to model more complex mathematical systems. 7.2.1 Rings Definition and Basic Properties A ring is an algebraic structure consisting of a set together with two binary operations, usually denoted as addition (+) and multiplication (·), satisfying the following axioms: is an abelian group: for all (addition is commutative) for all (addition is associative) There exists an element such that for all (additive identity) For each , there exists an element such that (additive inverse) is a semigroup: for all (multiplication is associative) The distributive laws hold: for all (left distributivity) for all (right distributivity) Types of Rings Commutative Ring: A ring is commutative if multiplication is commutative, i.e., for all . Ring with Unity: A ring has a multiplicative identity (or unity) if there exists an element such that for all . Integral Domain: A commutative ring with unity is an integral domain if it has no zero divisors, i.e., if implies either or . Examples of Rings The set of integers with standard addition and multiplication forms a commutative ring with unity. The set of even integers with standard addition and multiplication forms a commutative ring without unity. The set of matrices over a ring forms a ring under matrix addition and multiplication. This ring is non-commutative for . The set of polynomials with coefficients from a ring forms a ring under polynomial addition and multiplication. 7.2.2 Fields Definition and Basic Properties A field is a commutative ring with unity in which every non-zero element has a multiplicative inverse. Formally, a field is a set with two binary operations, addition (+) and multiplication (·), satisfying the following axioms: is an abelian group with identity element 0. is an abelian group with identity element 1. The distributive laws hold: for all for all Examples of Fields The set of rational numbers with standard addition and multiplication. The set of real numbers with standard addition and multiplication. The set of complex numbers with standard addition and multiplication. Finite fields where is a prime number, consisting of the integers modulo . The field of rational functions over a field . 7.2.3 Subrings and Subfields A subring of a ring is a subset that is itself a ring under the operations of . For to be a subring, it must: Be closed under addition and multiplication Contain the additive identity 0 Contain the additive inverse of each of its elements Similarly, a subfield of a field is a subset that is itself a field under the operations of . 7.2.4 Ring Homomorphisms and Isomorphisms A ring homomorphism is a function between rings that preserves the ring operations: for all for all If has a unity and has a unity , we often require that . A ring homomorphism that is bijective is called a ring isomorphism. Two rings are isomorphic if there exists an isomorphism between them. 7.2.5 Ideals and Quotient Rings An ideal of a ring is a subset that satisfies: is a subgroup of For all and , both and Given an ideal of a ring , we can form the quotient ring , whose elements are the cosets for . 7.2.6 Field Extensions A field extension is a pair of fields and such that is a subfield of . The degree of the extension, denoted , is the dimension of as a vector space over . Algebraic Extensions An element is algebraic over if there exists a non-zero polynomial such that . If every element of is algebraic over , then is called an algebraic extension. Transcendental Extensions An element is transcendental over if it is not algebraic over . If contains at least one transcendental element, it is called a transcendental extension. 7.2.7 Applications of Rings and Fields Rings and fields have numerous applications across mathematics: Number Theory: The study of algebraic number fields and their rings of integers. Algebraic Geometry: The study of polynomial rings and their quotients. Coding Theory: Finite fields are used in the construction of error-correcting codes. Cryptography: Field arithmetic is fundamental to many cryptographic protocols. Physics: Field theory provides a mathematical foundation for quantum field theory. Exercises 1. Prove that the set of all matrices with real entries forms a ring that is not commutative. 2. Show that the set of all polynomials with rational coefficients forms an integral domain but not a field. 3. Prove that if is a ring with unity and is a subring containing the unity, then is a ring with unity. 4. Let be a commutative ring. Prove that the set of all polynomials over forms a commutative ring. 5. Show that in a field, every non-zero element has exactly one multiplicative inverse. 6. Prove that a finite integral domain is a field. 7. Let be a field and the ring of polynomials over . Show that is not a field. 8. Determine all the ideals of the ring of integers. 9. Prove that the quotient ring is a field if and only if is prime. 10. Let be a field and an irreducible polynomial. Show that is a field.