When answering questions 3 and 4, What would be necessary to prove #2 and #3 are true? Outline a suggested proof for each of these. FieldsDefinition: An algebraic system {S, +,· } consisting of a set S together with twooperations + and , is called a field if it has the following properties.Va, b, c in S:Al.Addition is associative: a + (b + c) = (a + b) +cA2.Addition is commutative: a + b=b+aАЗ.Zero: 3 an element 0 in S such that a +0 = aA4.Opposite: 3 an clement -a such that a +-a = 0M1. Multiplication is associative: a(bc) = (ab)cM2. Multiplication is commutative: ab = baM3.One: 3 an element 1 in S such that la = aM4. Reciprocal: if a + 0,3 an clement - such that a-1= 1aMultiplication is distributive over addition: a (b + c) = ab + acD.1. Explain why the integers with + and · are not a field.2. Explain why the rational numbers with + and - are a field.3. Show that the set of numbers mod 5 with e and ® is a field.4. Show that the set of numbers mod 6 with e and © is not a field.

Name: When answering questions 3 and 4, What would be necessary to prove #2
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Description: When answering questions 3 and 4, What would be necessary to prove #2 and #3 are true? Outline a suggested proof for each of these. FieldsDefinition: An algebraic system {S, +,· } consisting of a set S together with twooperations + and , is called a

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2. Explain why the rational numbers with + and - are a field.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.7: Introduction To Coding Theory (optional)

Problem 12E: Suppose that the check digit is computed as described in Example . Prove that transposition errors...

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