How do I determine if a field F is the quotient field of an integral domain

Question

My question states: "In each case, determine if the field F is the quotient field of integral domain D (up to isomorphism)"

For example, if I have $D=\mathbb{Z}_7, F=\mathbb{Z}_{14}$, how would I go about showing this?

Another example would be $D=\mathbb{R},F=\mathbb{C}$

Any help appreciated

Answer 1

Roughly speaking, the quotient field of an integral domain $D$ is the "smallest" field $F$ which contains $D.$

For example, if $D=\mathbb{Z},$ then $F=\mathbb{Q}$ because any field containing $\mathbb{Z}$ has to contain all the fractions $a/b$ where $a,b \in \mathbb{Z}$ with $b\neq 0.$

In both your examples ($D=\mathbb{Z}_7$ and $D=\mathbb{R}$), the integral domain $D$ is already a field and so the smallest field containing $D$ is actually $D$ itself.

As an extra exercise... Describe the quotient field of the following integral domain: $$\mathbb{Z}[i]:=\{a+bi \in \mathbb{C} \, : \, a,b \in \mathbb{Z}\}.$$ Remember: the idea is to take your integral domain $D$ and throw in all the fractions of elements in $D.$