Casus irreducibilis

In algebra, casus irreducibilis (Latin for "the irreducible case") is one of the cases that may arise in attempting to solve a cubic equation with integer coefficients with roots that are expressed with radicals. Specifically, if a cubic polynomial is irreducible over the rational numbers and has three real roots, then in order to express the roots with radicals, one must introduce complex-valued expressions, even though the resulting expressions are ultimately real-valued. This was proved by Pierre Wantzel in 1843.

One can decide whether a given irreducible cubic polynomial is in casus irreducibilis using the discriminant D, via Cardano's formula. Let the cubic equation be given by

Then the discriminant $D$ appearing in the algebraic solution is given by

More generally, suppose that F is a formally real field, and that p(x) ∈ F[x] is a cubic polynomial, irreducible over F, but having three real roots (roots in the real closure of F). Then casus irreducibilis states that it is impossible to find any solution of p(x) = 0 by real radicals.

To prove this, note that the discriminant $D$ is positive. Form the field extension $F (\sqrt D)$ . Since this is $F$ or a quadratic extension of $F$ (depending in whether or not $D$ is a square in $F$ ), $p (x)$ remains irreducible in it. Consequently, the Galois group of $p (x)$ over $F (\sqrt D)$ is the cyclic group $C 3$ . Suppose that $p (x) = 0$ can be solved by real radicals. Then $p (x)$ can be split by a tower of cyclic extensions

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