An algebraic number is an algebraic integer if it is a polynomial root of a monic polynomial with integer coefficients:Equivalently, its algebraic minimal polynomial over has integer coefficients. The requirement that the leading coefficient be one matters: satisfies and is an algebraic number, but is not an algebraic integer. In fact the rational numbers that are algebraic integers are exactly the integers.[ref]
Some useful computed examples are:
- satisfies .
- satisfies .
- satisfies , although its displayed coefficients in the basis are not integers.
- has algebraic minimal polynomial , so it is not an algebraic integer even though it satisfies .
Algebraic integers are closed under addition, subtraction, and multiplication. Thus they provide an arithmetic ring, while division is allowed only when the result is again an algebraic integer. The algebraic integer module criterion explains this closure without having to find a new polynomial for every sum and product.
An **algebraic integer** is a special type of number in algebraic number theory. It is defined as a complex number that is a root of a monic polynomial (a polynomial whose leading coefficient is 1) with integer coefficients.
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