Casus Irreducibilis

Casus irreducibilis is the case of a cubic polynomial with rational coefficients that is irreducible over and has three real roots. No root can then be expressed from the rational numbers using only addition, subtraction, multiplication, division, and real radicals (Dummit and Foote 2004, pp. 625-639). Nonreal intermediate quantities are unavoidable in a radical expression, even though the result is real.

More generally, let be an irreducible polynomial that is also a real-rooted polynomial. If any root of can be expressed from using real radicals, then its polynomial degree must be for some nonnegative integer (Isaacs 1985, Isaacs and Moulton 1998).

For example, Trott (2006) considers the cubic equation

(1)

Its polynomial discriminant is , so it has three distinct real roots. It has no rational root and is therefore irreducible over . One root is

(2)

where the cube roots denote principal roots. The two summands are complex conjugates, so their sum is real. All three roots can instead be written using trigonometric functions as

(3)

for , 1, 2. This form uses real cosine and inverse cosine values and does not express the roots using real radicals.