# Thread: Intuition on Integral Over

1. Could somebody write me the intuition behind the concept of "Integral Over"? Please do not write me its formal definition, I can easily get it from textbook. What I am also looking for is its motivation behind it. Please give me also examples.

For your convenience, the formal definition according to goes like this:

In commutative algebra, an element $b$ of a commutative ring $B$ is said to be integral over $A$, a subring of $B$, if there are $n â‰¥ 1$ and $a_{j}\ in A$ such that

$$b^{n}+a_{n-1}b^{n-1}+\cdots +a_{1}b+a_{0}=0.$$

That is to say, $b$ is a root of a monic polynomial over $A$. If every element of $B$ is integral over $A$, then it is said that $B$ is integral over $A$, or equivalently $B$ is an integral extension of $A$.

I understand that plain simple English definition runs the risk of imprecision; I will take it as working definition only to be improved as I progress along. Thank you for your times and gracious helping hand.

2. The motivating example comes from the ring of real numbers. You probably know that every real number is either algebraic (like $\sqrt2$) or transcendental (like $\pi$). The algebraic numbers are those that satisfy a polynomial equation with rational coefficients. So they are exactly the real numbers that are integral over $\Bbb Q$. The set $\Bbb A$ of all algebraic numbers is a ring (in fact, a field – see ), and the ring $\Bbb A$ is an integral extension of $\Bbb Q$.

The motivating example comes from the ring of real numbers. You probably know that every real number is either algebraic (like $\sqrt2$) or transcendental (like $\pi$). The algebraic numbers are those that satisfy a polynomial equation with rational coefficients. So they are exactly the real numbers that are integral over $\Bbb Q$. The set $\Bbb A$ of all algebraic numbers is a ring (in fact, a field – see ), and the ring $\Bbb A$ is an integral extension of $\Bbb Q$.