Roots Appear in Quanta

We start with a special case. Consider an irreducible quintic polynomial f(X) = X + a1X + a2X + a3X + a4X + a5 with rational coefficients and with three real roots and one pair of complex conjugate roots. For example, f(X) could be X − 10X + 5. Question. If α is a root of f , then how many roots of f lie in the field Q(α)? The field Q(α) is obtained by adjoining the root α to Q. Thus Q(α) contains at least one root of f , and of course it can contain at most five roots of f . Answer. The number r(f) of roots of f in Q(α) is 1. We prove that, for an arbitrary irreducible polynomial f and root α, r(f) divides the degree of f . For the quintic under discussion, adjoining one of the real roots cannot possibly produce the nonreal roots, so r(f), being a divisor of 5, must be 1. An informal survey of books and colleagues indicates that the divisibility result “r(f) divides the degree” is not well known. In what follows, K is a field and, unless stated otherwise, all roots and field extensions are taken in a fixed algebraic closure K of K. When K = Q, we always take K inside the complex numbers so that we can speak of real roots and nonreal roots. Theorem 1. Let f(X) in K[X] be an irreducible polynomial, and let α be a root of f . Set