# Splitting it every which Way

Algebra Level 5

Consider a cubic polynomial $$f(x)$$ with integer coefficients that has three distinct complex roots $$a,b,c$$. What are the possible degrees of $$\mathbb{Q}(a,b,c)$$ over $$\mathbb{Q}$$? Enter the sum of those possible degrees.

Clarification:

$$\mathbb{Q}(a,b,c)$$ consists of all complex numbers of the form $$g(a,b,c)$$ where $$g(x,y,z)$$ is a polynomial with rational coefficients. The degree of $$\mathbb{Q}(a,b,c)$$ is its dimension as a vector space over $$\mathbb{Q}$$.

Relevant Wiki: Algebraic number theory.

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