The factoring of polynomials
This article is about the factorization of polynomials:
First, I’d like to discuss the most important trick that is used directly or implicitly in most of the theorems given below. Let us consider the polynomial . Let be the first coefficient, starting from , not to be a multiple of in . Let be the first such coefficient in . Then the coefficient of in where is definitely a multiple of , the coefficient of is definitely NOT a multiple of , and the coefficient of where , we can’t say anything about, except for the coefficient of which is definitely not a multiple of .
1. Gauss’s lemma (polynomial)- A primitive polynomial is one one with integer coefficients such that the gcd of all the coefficients is 1. The lemma states that if a primitive polynomial is irreducible over , then it is irreducible over . In other words, if a primitive polynomial has a root, it has to be an integer. The converse is also obviously true, as . A very interesting proof can be found here, although Proof 1 is not completely correct. The characterization of should actually be . This is to include the possibility of or . Also, note that in all other coefficients , we are unlikely to find a glaring contradiction (that doesn’t divide this coefficient). This proof by explicit demonstration is indeed brilliant. But wait. This just proves that the product of primitive polynomials is primitive. It doesn’t prove that a primitive polynomial can only be factored into primitive polynomials. This proves the aforementioned. The most important thing to concentrate on here is the how to convert rational polynomials into a constant times a primitive polynomial. Any irreducible polynomial can be converted into a constant times a product of two primitive polynomials. It is only when the original polynomial is also primitive that this constant is .
2. Rational root theorem- It states that if a polynomial with integer coefficients has a rational root , then divides the constant and divides the leading coefficient (if the polynomial is , then and ). A proof is given here. I’d like to add certain things to the proof given in the article for a clearer exposition. First of all, by taking out the integral gcd of the coefficients, make the polynomial primitive. Let the gcd be . Deal with this primitive polynomial. Using Gauss’s lemma and the proof in the article, we can easily deduce that and . This both these are true, and as is an integer, we get and .
3. Eisenstein criterion- The statement and the proof are given here. I want to discuss the true essence of the proof. The most important condition is that . What this essentially does is it splits between and ; ie it can’t divide both. How does the splitting of change anything? We’ll come to that. Another important condition is that . This forces us to conclude that not all or are divisible by . Hence, there is a first element and a which are not divisible by . If , then is not divisible by , which contradicts the third condition that for . How? Because .All terms except are divisible by . This is where the splitting helped. If there was no such splitting (if ) then could have been divisible by ). Similarly, if , then . Remember the point that I elaborated at the very beginning of the article? Try to correlate that with this proof. Here becomes or , as the first coefficient not to be divisible by is or .