An elementary symmetric polynomial question

Question

If we have three complex numbers $a,b,c$ such that the three elementary symmetric polynomials $a+b+c$, $ab+ac+bc$, and $abc$ are all integers, what characteristics can one deduce about $a,b,c$? For example, must they all be rational? Or even integral?

Answer 1

Let $p=a+b+c$, $q=ab+bc+ca$, and $r=abc$. Then $a$, $b$, and $c$ are the roots of the polynomial equation $x^3-px^2+qx-r$. Certainly the roots of such a polynomial equation need not be integers. Take for instance the polynomial $x^3-2$.

But we can say a few things. For example, by the Rational Roots Theorem, any rational root of the cubic must be an integer.

We can say that at least one of $a,b,c$ is real, since every cubic polynomial with real coefficients has a real root.

Answer 2

They are algebraic integers. The fields ${\mathbb Q}(a)$, ${\mathbb Q}(b)$, and ${\mathbb Q}(c)$ are each of degree at most $3$ over ${\mathbb Q}$. The field ${\mathbb Q}(a,b,c)$ is of degree at most 6 over ${\mathbb Q}$.