The analogy between integers and polynomials

Integers and polynomials form number systems which have the following things in common:

  • The two number systems closed under addition, subtraction and multiplication.
  • Objects in the respective number systems can be factored into “primes.” You’re familiar with what an prime integer is. A prime polynomial is just a polynomial that doesn’t factor further. To make this more precise we have to specify what sorts of coefficients the factors can have.The simplest case is if the factors are allowed to have complex coefficients. In this case, according to the Fundamental Theorem of Algebra and the factor theorem a polynomial with complex coefficients can be factored into a constant times linear factors; so the “prime polynomials” are just the linear polynomials (ax – b) where a and b are complex numbers.The next simplest case is the case where the factors are allowed to have real coefficients. In this case, using the fundamental theorem of algebra and the fact that the roots of a polynomial with real coefficients come in complex conjugate pairs one can prove that every polynomial with real coefficients can be factored into linear polynomials and irreducible quadratic polynomials with real coefficients, so the “prime polynomials” are the linear polynomials (ax – b) where a and b are real and the quadratic polynomials Ax^2 + Bx + C which have real coefficients and non-real roots.Things become more complicated if we require that the coefficients be rational numbers. Most polynomials with rational coefficients do not factor further into polynomials with rational coefficients; I mentioned this the fifth paragraph of my post titled The Rational Root Theorem. So in this case most polynomials are prime, though obviously there are exceptions (to find one just take two polynomials with rational coefficients and multiply them together!)
  • Both integers and polynomials obey unique prime factorization.One needs to be a little bit careful here. A straightforward point is that two factorizations are considered to be the same if they differ only in the order of the factors, e.g. (x – 1)(x + 2) is considered to be the same factorization as (x + 2)(x – 1). More subtly, two factorizations are considered to be the same if the factors differ by units. The units of a number system like the integers or the polynomials are the elements which have inverses that are also integers/polynomials. The units in the set of integers are 1 and -1 (the latter of which is its own inverse). The units in the set of polynomials are the nonzero constant polynomials (the inverse of f(x) = a  is g(x) = 1/a). So for example the factorization 15 =  (3)(5) is considered to be the same as 15 = (-3)(-5). Similarly the factorization of 2x^2 – 12x + 10 given by (2x – 2)(x – 5)
    is considered to be the same as (x – 1)(2x – 10). Once one makes these qualifications, an integer has only one factorization into primes numbers and a polynomial has only one factorization into prime polynomials.
  • Long division is possible.Given an integer ‘a’ and an integer ‘b’ one can write a = b*q + r in exactly one way with ‘r’ is between 0 and b. Here ‘q’ is called the “quotient” of ‘a’ upon division by ‘b’ and ‘r’  is called the remainder.Similarly, if f(x) is a polynomial and g(x) is a polynomial then one can write f(x) = g(x)q(x) + r(x)in exactly one way with the degree of r(x) between 0 and the degree of g(x); again, q(x) is called the quotient and r(x) is called the remainder.
  • Neither the integers nor the polynomials are closed under division but they have natural extensions that are closed under division: the smallest number system that the integers can be extended to which is closed under division is the rational numbers and the smallest number system that the polynomials can be extended to which is closed under division is the set of rational functions. Combining this with unique prime factorization one can say that every rational number/rational function factors uniquely into prime integers/polynomials (possibly with negative exponents to account for the denominators).
  • Both rational numbers and rational functions have partial fraction decompositions. In class we talked about how to find the partial fraction decomposition of something like f(x) = 1/[(x – 1)(x + 2)] by writing it as A/(x – 1) + B/(x +2) and solving for A and B.More generally, given a rational function f(x)/g(x) with f(x) having smaller degree than g(x), one can factor g(x) into prime polynomials and write f(x)/g(x) as a sum of terms each one of which has denominator consisting of only a single prime polynomial (possibly raise to a power). This is covered thoroughly in Section 11.6 of the text.What we didn’t talk about in class is how one can write a number like 1/15 as a sum of two terms rational number search with denominator a prime factor of 15. We try to write1/15 = x/3 + y/5.where ‘x’ and ‘y’ are integers. As in the case of rational functions, we clear the denominators to get 1 = 5x + 3y. The question now is how to find ‘x’ and ‘y’ that work. In contrast with the case of rational functions the situation here is more subtle: the methods that we discussed in class that work for rational functions don’t work here. In this particular case we can eyeball the solution x = 2 and y = -3 to decompose 1/15 as1/15 = 2/3 – 3/5but this evades the general issue: suppose that n is a product of two primes p and q and that we want to write 1/n = x/p + y/q for some integers a and b: is this possible and if so how do we find ‘a’ and ‘b’ that work?

    Equivalently, given primes p and q how can we find integers x and y such that

    1 = px + qy ?

    The answer to this question was known to the ancient Greeks: it comes from the so-called Euclidean algorithm. There’s much to say here but I’ll refrain from doing so now; postponing the discussion for later. Instead I’ll just remark that the fact that it’s more difficult to prove a statement about integers than about polynomials is in fact typical in mathematics. For example, Andrew Wiles proved Fermat’s Last Theorem around 1995 but the the polynomial analog had already been proven around 1900.

    ————–

    A book that I like which expands on some of the themes that I mention above is Ronald Irving’s Integers, Polynomials and Rings. The analogy between integers and polynomials runs still deeper than what I have discussed here but we’ve reached a suitable breaking point.  I hope this will be useful for you in organizing some of what we’ve been doing in class.

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