In my textbook there's a theorem that goes like this:
Let $D$ be an integral domain. Then $D$ can be embedded in a field of fractions $F_D$, where any element in $F_D$ can be expressed as the quotient of two elements in $D$. Furthermore, the field of fractions $F_D$ is unique in the sense that if $E$ is any field containing $D$, then there exists a map $\phi : F_D \rightarrow E$ giving an isomorphism with a subfield of $E$ such that $\phi (a) = a$ for all elements $a\in D$.
I don't understand what it means to say that $D$ can be embedded in the field of fractions. I just don't get what the word "embedded" means, my book doesn't define it!
And what does it mean to say that the field of fractions $F_D$ is unique? What are they getting at by saying that?
Also, I don't understand the "significance" of this theorem. It seems very random and out of the blue to me. Can anyone enlighten me?
Sorry if this is a stupid question, but I really don't understand this at the moment, despite having tried and tried for hours. What makes it worse is that I love abstract algebra and really want to master these concepts but at the moment I am not!