This is a problem I've been working on. can you guys point out some uniqueness theorems you'd find helpful or an example proof? I'm pretty lost.
2026-04-08 23:04:53.1775689493
Integral domain and homomorphisms
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Every element of $F_D$ can be written as $x/y$ with $x,y\in D$ and $y\ne0$ (but not uniquely so, beware!).
First prove that $F_\theta(x/y) = \theta(x)/\theta(y)$ is a well-defined field homomorphism.
Next prove that every field homomorphism $F_D \to K$ extending $\theta$ must send $x/y$ to $\theta(x)/\theta(y)$ and so must coincide with $F_\theta$.
It's all pretty straightforward really. Give it a try.