[Math] Prove the fractional field of an integral domain is the smallest field containing the integral domain

abstract-algebrafield-theoryintegral-domain

I have two questions about the fractional field of an integral domain.

Given an integral domain $D$:

  1. Is there a difference between saying "the fractional field of $D$ is the smallest field containing $D$" or "the fractional field of $D$ is the smallest field containing an embedding of $D$"?

  2. How do you prove that the fractional field is the smallest field containing $D$ (or an embedding of $D$, if there is a difference…)? Specifically, I want to show that if $F$ is any field containing $D$, then $F$ must contain the fractional field of $D$.

Thanks for your help.

Best Answer

The right (i.e. categorical) way to say this (without the ambiguities of words like "smallest", "containing", etc.) ought to be that the inclusion $\iota: D\to Q(D)$ has the following universal property:

If $K$ is a field, and $f: D\to K$ is any morphism of rings, then there is a unique morphism of fields $g : Q(D) \to K$ such that $f = g \circ \iota$.

(In particular, $Q(D)$ embeds into any field that $D$ embeds into.)

This property uniquely determines (up to isomorphism) not only $Q(D)$, but $\iota$ as well.

And it's easily proved, since $g(1/b)g(b)=g(1)$ forces $g(a/b) = f(a)/f(b)$, so this amounts to checking that $a/b \mapsto f(a)/f(b)$ is actually a homomorphism.

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