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.


\(\mathbb{Q}(a,b,c)\) 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.

A complete solution will contain examples of all possible cases.


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