Splitting fields for $x^3-3$ and $x^5-1$

Question

I'm looking for the splitting fields of

(a) $x^3-3$

(b) $x^5-1$.

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EDIT:

(a) Thanks to all the hints and suggestions, the three roots are

$x_1=3^{\frac{1}{3}}$, $x_2=e^{\frac{2 \pi i}{3}}3^{\frac{1}{3}}$, $x_3=e^{\frac{4 \pi i}{3}}3^{\frac{1}{3}}$

Now, the question doesn't specify the field over which these polynomials are defined, I'll take a guess and say $Q$. Now, all the roots can be generated from $x_2=e^{\frac{2 \pi i}{3}}3^{\frac{1}{3}}$ over the rationals, so is the answer $Q(e^{\frac{2 \pi i}{3}}3^{\frac{1}{3}})$ correct?

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(b) Again, the roots are the 5 complex roots of unity, all of which can be generated by the root $x_1=e^{\frac{2 \pi i}{5}}$. So would the correct answer now be $Q(e^{\frac{2 \pi i}{5}})$

Answer 1

Hints:

• A real number has $3$ cube roots in $\mathbf C$. One is the standard real cube root, he other two are this real cube root, multiplied by one of the complex cube roots of unity.
• For $x^5-1$, solve it in the form $\mathrm e^{i\theta}$.