Consider integral domains $A$ and $B$ and their field of fractions $F(A)$ and $F(B)$. According to wikipedia, if $B$ is the integral closure of $A$ in an algebraic closure $L$ of $F(A)$, then $L=F(B)$.
Question 1: Which book states this result? I need a reference for this result.
Question 2: Is the converse true? When $F(B)$ is the algebraic closure of $F(A)$, is true that $B$ is the integral closure of $A$? If the answer is yes, then a reference of the proof will also be considered an answer.
It has been difficult to find a good reference so I need a hand with this. Thanks.
For qstn 1 pick an element in F(B). This element is of the form a/b where a,b are algebraic over F(A) Hence a/b is in L. Now pick an element l in L. Consider the minimal polynomial of the element in L. Multiply by suitable constant to get the element cl which is integral over A. one choice of c would be (coefficient of highest degree term * product of denominators of all the nonzero coefficients)^degree of minimal polynomial. The c u chose belongs to A. and cl belongs to B. Thus l belongs to F(B).
As for the second it is not true. Just adjoin another element from the algebraic closure to the integral closure.