Polynomials Over The Rational Field

We specialize the general discussion to that of polynomials whose coefficients are rational numbers. Most of the time the coefficients will actually be integers. For such polynomials we shall be concerned with their irreducibility.

DEFINITION: The polynomial where the are integers is said to be primitive if the greatest common divisor of is 1.

LEMMA 3.10.1: If and are primitive polynomials, then is a primitive polynomial.


DEFINITION: The content of the polynomial where the 's are integers, is the greatest common divisor of the integers

Clearly, given any polynomial with integer coefficients it can be written as where is the content of and where is a primitive polynomial.

THEOREM 3.10.1 (Gauss' Lemma): If the primitive polynomial can be factored as the product of two polynomials having rational coefficients, it can be factored as the product of two polynomials having integer coefficients.


DEFINITION: A polynomial is said to be integer monic if all its coefficients are integers and its highest coefficient is 1.

Thus an integer monic polynomial is merely one of the form where the 's are integers. Clearly an integer monic polynomial is primitive.

COROLLARY: If an integer monic polynomial factors as the product of two non-constant polynomials having rational coefficients then it factors as the product of two integer monic polynomials.

Proof: We leave the proof of the corollary as an exercise for the reader.

The question of deciding whether a given polynomial is irreducible or not can be a difficult and laborious one. Few criteria exist which declare that a given polynomial is or is not irreducible. One of these few is the following result:

THEOREM 3.10.2 (The Eisenstein Criterion): Let be a polynomial with integer coefficients. Suppose that for some prime number



Then
is irreducible over the rationals.