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  1. MHB Apprentice

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    #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 $n ≥ 1$ 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.

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    Opalg's Avatar
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    #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 $\Bbb Q$.

  3. MHB Apprentice

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    Quote Originally Posted by Opalg View Post
    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 $\Bbb Q$.
    Thank you for your gracious helping hand, apologize for getting back to you late. ~MA

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