where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients.
All algebraic integers are therefore algebraic numbers, but it can be shown that not all algebraic numbers are algebraic integers.
One may show that if P(x) is a non-monic primitive polynomial with integer coefficients that is irreducible over Q, then none of the roots of P are algebraic integers. Here the word primitive means that coefficients of P are coprime (i.e. the greatest common divisor of the set of coefficients of P is 1; note that this is weaker than requiring the coefficients to be pairwise relatively prime.)
The sum of two algebraic integers is an algebraic integer, and so is their difference; their product is too, but not necessarily their ratio. An integer root of an algebraic integer is also an algebraic integer. So all radical integers are algebraic integers but not all algebraic integers are radical integers. In other words, the algebraic integers form a ring that is closed under the operation of extraction of roots.
The term rational integer is used, in algebraic number theory, to distinguish these 'ordinary' integers, in the rational numbers, from other concepts such as the Gaussian integers.
The integer q is called the quotient and r is called the remainder, resulting from division of a by b.
Integer datatypes are typically implemented using a fixed number of bits, and even variable-length representations eventually run out of storage space when trying to represent especially large numbers.
If an algebraic number satisfies such an equation as given above with a polynomial of degree n and not such an equation with a lower degree, then the number is said to be an algebraic number of degree n.
In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.
The name algebraicinteger comes from the fact that the only rational numbers which are algebraicintegers are the integers, and because the algebraicintegers in any number field are in many ways analogous to the integers.
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