Please answer the following questions. All statements must be explicitly justified.

Introduction: Fields are an Important algebraic structure. A few examples of fields are real and rational numbers, as well as modular numbers if the modulus is a prime number. Requirements: Your submission must be your originai work. No more than a combined totai of 30% of the submission and no more than a 10% match to any one individuai source can be directi'y quoted or cioseiy paraphrased from sources, even if cited correctiy. Use the Turnitin Originai'ity Report availabie in Taskstream as a guide for this measure of originaiity. You must use the rubric to direct the creation of your submission because it provides detailed criteria that wiii be used to evaiuate your work. Each requirement beiow may be evaiuated by more than one rubric aspect. The rubric aspect titles may contain byperiinks to reievant portions of the course. Given: An Integral domain 2 Is a ring for the operations + and * with three additional properties: 1. The commutative property of *: For any elements x and yin Z, x*y=y'x. 2. The unity property: There is an element 1 in 2 that is the identity for *, meaning for any 2 In 2, 2*1=z. Also, 1 has to be shown to be different from the identity of +. 3. The no zero divisors property: For any two elements a and b In 2 both different from the identity of +, a*b$0. A field F Is an Integral domain with the additional property that for every element x in Fthat is not the Identity under +, there is an element y in F so that x*y=1 (1 is notation for the unity of an integral domain). The element y is called the multiplicative Inverse of x. Another way to explain this property Is that multiplicative inverses exist for every nonzero element. Modular multiplication, [*2], Is defined in terms ofinteger multiplication by this rule: [a],.I1 [*] [b],,.| = [a * bjm Note: For ease of writing notation, foifow the convention of using just plain ' to represent both ['1' and *. Be aware that one symbol can be used to represent two different operations (moduiar muitipiication versus integer muitipiication). A. Prove that the ring 231 {integers mod 31) Is an integral domain by using the definitions given above to prove the following are true: 1. The commutative property of [*] 2. The unity property 3. The no zero divisors property B. Prove that the Integral domain 23] (Integers mod 31) Is a field by using the definition given above to prove the existence of a multiplicative Inverse for every nonzero element. C. Acknowledge sources, using ARA-formatted ln-text citations and references, for content that is quoted, paraphrased, or summarized