What is an integral element?

Question

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?

Answer 1

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.

Answer 2

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 $x-\frac{1}{2}$.

Answer 3

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.