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In mathematics, an integer-valued polynomial (also known as a numerical polynomial) () is a polynomial whose value () is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true. For example, the polynomial
A constant coefficient, also known as constant term or simply constant, is a quantity either implicitly attached to the zeroth power of a variable or not attached to other variables in an expression; for example, the constant coefficients of the expressions above are the number 3 and the parameter c, involved in 3=c ⋅ x 0.
Similarly, an integer polynomial is a polynomial with integer coefficients, and a complex polynomial is a polynomial with complex coefficients. A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial.
This results from the rational root theorem, which asserts that, if the rational number is a root of a polynomial with integer coefficients, then q is a divisor of the leading coefficient; so, if the polynomial is monic, then =, and the number is an integer.
The larger points come from polynomials with smaller integer coefficients. If a polynomial with rational coefficients is multiplied through by the least common denominator, the resulting polynomial with integer coefficients has the same roots. This shows that an algebraic number can be equivalently defined as a root of a polynomial with either ...
The value y = a n x is an algebraic integer because it is a root of q(y) = a n − 1 n p(y /a n), where q(y) is a monic polynomial with integer coefficients. If x is an algebraic number then it can be written as the ratio of an algebraic integer to a non-zero algebraic integer. In fact, the denominator can always be chosen to be a positive integer.
p is an integer factor of the constant term a 0, and; q is an integer factor of the leading coefficient a n. The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials. The integral root theorem is the special case of the rational root theorem when the leading coefficient is a n ...
From coefficients in an algebraic extension to coefficients in the ground field (see below). From rational coefficients to integer coefficients (see below). From integer coefficients to coefficients in a prime field with p elements, for a well chosen p (see below).