Automorphism of an integral domain extends to an automorphism of the quotient field

Question

Every automorphism of an integral domain can be extended to an automorphism of its quotient field.

Please help to start with the proof!!

Answer 1

Let $A$ be an integral domain and let $k$ be it's field of fractions and let $i : A \to k$ be the canonical inclusion. Let $\phi : A \to A$ be an automorphism. Then $i\phi : A \to k$ is a homomorphism with the property that for every non-zero $a \in A, i\phi(a)$ is a unit in $k$ and every element of $k$ can be written as $(i\phi(a))(i\phi(b))^{-1}$ for some $a, b \in A, b \neq 0.$ Then by the universal property of field of fractions, there exists an isomorphism $\Phi : k \to k$ such that $i\phi = \Phi i.$