Please answer number question 4 **Fields****Definition:** An algebraic system \(\{S, +, \cdot\}\) consisting of a set \(S\) together with two operations \(+\) and \(\cdot\), is called a *field* if it has the following properties.For all \(a, b, c\) in \(S\):- **A1.** Addition is associative: \(a + (b + c) = (a + b) + c\)- **A2.** Addition is commutative: \(a + b = b + a\)- **A3.** Zero: \(\exists\) an element \(0\) in \(S\) such that \(a + 0 = a\)- **A4.** Opposite: \(\exists\) an element \(-a\) such that \(a + (-a) = 0\)- **M1.** Multiplication is associative: \(a(bc) = (ab)c\)- **M2.** Multiplication is commutative: \(ab = ba\)- **M3.** One: \(\exists\) an element \(1\) in \(S\) such that \(1a = a\)- **M4.** Reciprocal: if \(a \neq 0\), \(\exists\) an element \(\frac{1}{a}\) such that \(a \cdot \frac{1}{a} = \frac{1}{a} \cdot a = 1\)- **D.** Multiplication is distributive over addition: \(a(b + c) = ab + ac\)1. Explain why the integers with \(+\) and \(\cdot\) are not a field.2. Explain why the rational numbers with \(+\) and \(\cdot\) are a field.3. Show that the set of numbers mod 5 with \(\oplus\) and \(\odot\) is a field.4. Show that the set of numbers mod 6 with \(\oplus\) and \(\odot\) is not a field.

Name: Please answer number question 4 **Fields****Definition:** An algebraic
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: Please answer number question 4 **Fields****Definition:** An algebraic system \(\{S, +, \cdot\}\) consisting of a set \(S\) together with two operations \(+\) and \(\cdot\), is called a *field* if it has the following properties.For all \(a, b, c\) i