Let $K$ be a field containing an integral domain $D$ and $F$ be the field of quotients of $D$. Then $K$ contains a field isomorphic to $F$.

Question

Let $K$ be a field containing an integral domain $D$ and $F$ be the field of quotients of $D$. Then $K$ contains a field isomorphic to $F$.

I have looked over some solutions but I don’t understand the general approach. That is, they start by defining a map $\phi:F \to K$, and show $\phi$ is an isomorphism. I understand what follows after that, but I am having trouble with the start of this approach.

Now, the main point of this problem is to show that $F$ is the smallest field containing D. And proving these kinds of problems involves assuming any field containing $F$ will be isomorphic to $F$ meaning that we can’t reduce $F$ any further.

Keeping this in mind, I am struggling with the following questions:

• Why does the approach highlighted above make sense?
• Why even phrase the question like this if we aren’t going to show an imaginary sub-field of $K$ and show it to be isomorphic to $F$?
• Is it possible to construct a proof with the following outline:

Let $F’$ be a field s.t. $F’ \subset K$, then show $F \cong F’$.

Answer 1

$F$ is the smallest field that $D$ can be embedded in. The field of fractions has the universal property that any embedding of an integral domain $D$ into a field $K$ extends to an embedding of $F$.