Abstract-Algebra-1 The Field of Quotients of an Integral Domain, guaranteed and verified 100% Pass

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The Field of Quotients of an Integral Domain
Some integral domains, such as ℤ, do not form fields. However, we can
always start with any integral domain and create a field from it by adding the
quotients of the non-zero elements. In the case of ℤ, that field is ℚ, the rational
numbers. In fact this construction gives the smallest field that contains the
original integral domain as a subdomain. This construction of a field of quotients
from a general integral domain, 𝐷, parallels the construction of ℚ from ℤ.

Let 𝐷 be an integral domain. We will create a field of quotients, 𝐹, that
contains 𝐷. The steps to create 𝐹 are:

1) Defining what the elements of 𝐹 are.
2) Defining the binary operations of addition and multiplication on 𝐹.
3) Checking that 𝐹 satisfies the field axioms.
In this construction think of 𝐷 as ℤ and 𝐹 as ℚ, even though this works for a
general integral domain 𝐷.



1. Let 𝐷 be an integral domain. Form 𝐷 × 𝐷 by 𝐷 × 𝐷 = {(𝑎, 𝑏)| 𝑎, 𝑏 ∈ 𝐷}.
𝑎 2
We think of (𝑎, 𝑏) as . If 𝐷 = ℤ, (2, 5) ∈ 𝐷 × 𝐷 = ℤ × ℤ represents .
𝑏 5


There’s a problem with this construction. We don’t want to divide by 0.

So define 𝑆 ⊆ 𝐷 × 𝐷 so that: 𝑆 = {(𝑎, 𝑏)⃒ 𝑎, 𝑏 ∈ 𝐷, 𝑏 ≠ 0}.
We still have a problem. For example, if 𝐷 = ℤ, (2, 3) and (4,6) both represent
the same rational number. To avoid this repetition we make the following
definition:

Def. Two elements (𝑎, 𝑏) and (𝑐, 𝑑) in 𝑆 are equivalent,

denoted (𝑎, 𝑏)~(𝑐, 𝑑) if, and only if, 𝑎𝑑 = 𝑏𝑐.