Proposition 5.15 on page 63 in Atiyah-Macdonald goes as follows:
Let $A \subset B$ be integral domains, $A$ integrally closed, and let $x \in B$ be integral over an ideal $ \mathfrak a$ of $A$. Then $x$ is algebraic over the field of fractions $K$ of $A$ and if its minimal polynomial over $K$ is $t^n + a_1 t^{n-1} + \dots + a_n$ then $a_1, \dots , a_n$ lie in $r(\mathfrak a)$.
According to Wikipedia, an algebraic element is defined as follows:
"If $L$ is a field extension of $K$, then an element a of $L$ is called an algebraic element over $K$, or just algebraic over $K$, if there exists some non-zero polynomial $g(x)$ with coefficients in $K$ such that $g(a)=0$."
1)Here $L$ is a field. Is it ok to call an element algebraic even if $L=B$ is just an integral domain? "algebraic" is never defined in AM.
2)Also, in the proof following the theorem: what is a conjugate of $x$?