What is an integral element? I'm trying to straighten out the definition of an integral element...
An integral element is not necessarily an integer itself, but is the root of a monic polynomial with integer coefficients? Does that sound right or am I off base?
 A: Let $A$ be a commutative ring and $B$ a subring of $A$. Then $a \in A$ is integral over $B$ if there is monic polynomial $f(x) \in B[x]$ such that $f(a)=0$. As I understand it, it is the same idea as an algebraic number $\alpha$ over a field $F$ except we are dealing with rings.
A: Yes, you're correct.
The set $\mathbb{Z}$ of integers forms a ring, i.e. an algebraic structure which allows you to make sense of addition and multiplication; then a real (or complex) number $r$ is integral if it satisfies a polynomial with integer coefficients and with leading coefficient $1$.
Generally, if $R$ and $S$ are commutative rings and $S \subseteq R$, then $r \in R$ is integral over $S$ if it is the root of a polynomial with coefficients in $S$ and with leading coefficient $1$. This is just the special case where $R = \mathbb{R}$ or $\mathbb{C}$ or whatever, and $S=\mathbb{Z}$.
For example, $\frac{1}{2}$ is not integral (over $\mathbb{Z}$) since by the rational root theorem if it is the root of a polynomial with integer coefficients then the leading coefficient cannot be $1$. However it is integral over $\mathbb{Q}$ since it is a root of the polynomial $x-\frac{1}{2}$.
A: Any number that is a root of a polynomial.. 
Its all depend on the polynomial structure.. For example. 
P(x) =X^2-7/6x+1/3.
Here  2/3 & 1/2 both are roots of this polynomial. 
Can we say that this example is valid for integral extension? where Q is subring of R. 
And can we say that 1/2 & 2/3 both are integral over R. 
