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Root(s) of a $3^{rd}$ degree polynomial over $ \Bbb Q$ are expressible using radicals with the imaginary $i$. If a root $r$ is real, by taking only the real part, $r$ is expressible using radicals over the rational numbers.

Why not? See here on Wikipedia.

Root(s) of a $3^\text{rd}$ degree polynomial over $ \Bbb Q$ are expressible using radicals with the imaginary $i$. If a root $r$ is real, by taking only the real part, $r$ is expressible using radicals over the rational numbers.

Why not? See Casus irreducibilis on Wikipedia.

Root(s) of a $3^{rd}$ degree polynomial over $ \Bbb Q$ are expressible using radicals with the imaginary $i$. If a root $r$ is real, by taking only the real part of $r$, it is expressible using radicals over the rational numbers.

Why not? See here on Wikipedia.

Root(s) of a $3^{rd}$ degree polynomial over $ \Bbb Q$ are expressible using radicals with the imaginary $i$. If a root $r$ is real, by taking only the real part, $r$ is expressible using radicals over the rational numbers.

Why not? See here on Wikipedia.

The minimal polynomial of $\cos(\pi/7)$ is $8x^3-4x^2-4x+1=0$ with three real roots. The degree is $<3$ so the roots are expressible using radicals with the imaginary $i$. Since the roots are real, taking only the real part, the roots are expressible using radicals of rational numbers.

Why not? See here on Wikipedia.

Root(s) of a $3^{rd}$ degree polynomial over $ \Bbb Q$ are expressible using radicals with the imaginary $i$. If a root $r$ is real, by taking only the real part of $r$, it is expressible using radicals over the rational numbers.

Why not? See here on Wikipedia.