Proposition: The field of fractions of an integral domain is a field
I founded the above proposition from Wikipedia.
"In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded" (http://en.wikipedia.org/wiki/Field_of_fractions)
How is this true?
Can you help me prove this one is true?
Best Answer
This will probably be easiest to see by the universal property that the field of fractions satisfies.
Let $R$ be an integral domain. The field of fractions can be described as a pair $(Frac(R),f)$, where $Frac(R)$ is a field, and $f:R\to Frac(R)$ an embedding, satisfying the following: if $k$ is another field and $g:R\to k$ is another embedding, then there exists an embedding $h:Frac(R)\to k$ such that the $g=h\circ f$.
From this definition, can you see why $Frac(R)$ is the smallest field containing $R$? Also, under this definition, $Frac(R)$ is automatically a field, because we define it to be so. We have yet to show that any such structure exists, but by the universal property, if it does exist, it is unique up to unique isomorphism. The usual construction of such an object mimics the construction of $\mathbb{Q}$ from $\mathbb{Z}$.