Galois Theory, 5th Edition

Chapter 3Factorisation of Polynomials

DOI: 10.1201/9781003213949-3

Not only is there an algebra of polynomials: there is an arithmetic. That is, there are notions analogous to the integer-based concepts of divisibility, primes, prime factorisation, and greatest common divisors. These notions are essential for any serious understanding of polynomial equations, and we develop them in this chapter.

Mathematicians noticed early on that if f is a product gh of polynomials of smaller degree, then the solutions of $f(t)=0$ are precisely those of $g(t)=0$ together with those of $h(t)=0$. For example, to solve the equation

$$\begin{array}{l}{t}^3-6t^2+11t-6=0\hfill \end{array}$$

we can spot the factorisation $(t-1)(t-2)(t-3)$ and deduce that the roots are $t=1,2,3$. From this simple idea emerged the ...