On the Impossibility of Doubling the Cube and Trisecting the Angle

As we saw in the last chapter, we can ask for factorizations of polynomials (into polynomials of lower degree), just as we can ask for possible factorizations of integers. In this context, we call a polynomial irreducible if it does not allow such a factorization. In other words, the irreducible polynomials play the role of prime numbers in the ring of polynomials. Every linear polynomial \(X - a\) must be irreducible, because already for reasons of degree it cannot possess a factorization into polynomials of lower degree. Due to the fundamental theorem of algebra, the linear polynomials are in turn the only irreducible ones, if we assume the algebraic numbers as the ring of coefficients. Over the rational numbers, the theory is more complicated, but also more interesting. Here, there are non-irreducible polynomials of higher degree like \(X^2 - 1 = (X - 1) \cdot (X + 1)\) , as well as irreducible ones like \(X^2 + 1\) . Given this, a natural question is how to determine whether a polynomial, say over the rational numbers, is irreducible. To this end, we provide a numerical method by which we can definitely determine whether such a polynomial is irreducible or not. This implies that every polynomial has an essentially unique decomposition into a product of irreducible polynomials, comparable to the prime factor decomposition of integers. Even though the numerical method we provide always works, there are simpler criteria for irreducibility for many cases. We present the Eisenstein criterion and another method that is based on irreducibility modulo a prime number. In particular, we compare the irreducibility of polynomials with rational coefficients with the irreducibility of polynomials with integer coefficients. An algebraic number z is defined as a complex number that is a root of a polynomial with rational coefficients. It turns out that among these there is exactly one irreducible normalized one, the so-called minimal polynomial of z. We simply call the degree of this minimal polynomial the degree of z. With the help of this degree concept, we finally show at the end of the chapter that neither the doubling of the cube nor the trisection of the angle is possible with compass and straightedge alone.