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
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.
Proof: Let and Suppose that the lemma was false; then all the coefficients of would be divisible by some integer larger than 1, hence by some prime number Since is primitive, does not divide some coefficient Let be the first coefficient of which does not divide. Similarly let be the first coefficient of which does not divide. In the coefficient of is
Now by our choice of so that
Similarly, by our choice of so that
By assumption, Thus by (1), which is nonsense since and This proves the lemma.
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.
Proof: Suppose that where and have rational coefficients. By clearing of denominators and taking out common factors we can then write
where and are integers and where both and have integer coefficients and are primitive. Thus
The content of the left-hand side is since is primitive; since both and are primitive, by Lemma 3.10.1 is primitive, so that the content of the right-hand side is Therefore and where and have integer coefficients. This is the assertion of the theorem.
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.
Proof: Without loss of generality we may assume that is primitive, for taking out the greatest common factor of its coefficients does not disturb the hypotheses, since If factors as a product of two rational polynomials, by Gauss' lemma it factors as the product of two polynomials having integer coefficients. Thus if we assume that is reducible, then
where the 's and 's are integers and where and Reading off the coefficients we first get Since must divide one of or Since cannot divide both and Suppose that Not all the coefficients can be divisible by ; otherwise all the coefficients of would be divisible by which is manifestly false since Let be the first not divisible by Thus and the earlier 's. But
so that However, which conflicts with This contradiction proves that we could not have factored and so is indeed irreducible.