Roots appear in quanta

When you construct a field extension by adjoining one root of an irreducible polynomial, how many roots of that polynomial appear in the constructed field? That number is called the root quantum number. It places restrictions on how many roots appear in arbitrary field extensions, and it places restrictions on how a polynomial can factor over a field extension. Connections with Galois theory are given, and exercises at the end give a complete classification of the possible root quantum numbers for polynomials with rational coefficients.