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I'm looking for the splitting fields of

(a) $x^3-3$

(b) $x^5-1$.

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?

(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 $Q(e^{\frac{2 \pi i}{5}})$

Mike
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  • Splitting fields over which ground field? – Torsten Schoeneberg Dec 02 '18 at 23:45
  • @TorstenSchoeneberg The question doesn't specify, perhaps there is an obvious choice? Most of the relevant section is concerned with extensions over the rationals, so my safe assumption is the rationals. – Mike Dec 02 '18 at 23:46
  • Also, be very careful with writing a negative number to the power of a fractional exponent. That is an ill-defined expression and that probably caused part of the problem here. Cf. https://math.stackexchange.com/q/317528/96384 – Torsten Schoeneberg Dec 02 '18 at 23:49
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    Craig, you should have a look first at what Mathematica means by $(-3)^{1/3}$. – Jean-Claude Arbaut Dec 02 '18 at 23:54
  • Thanks, let me rewrite the roots through euler's formula, which is what mathematica means, and edit this question. – Mike Dec 02 '18 at 23:56
  • When you do that, you will basically use the $n$th roots of unity. A primitive root of unity is any root $\omega$ that can generate all the others by taking successive powers $\omega^m$. The roots are given by $\exp (2ik\pi/n)$ with $0\le k<n$, and you can prove that the primitive ones are such that $gcd(k,n)=1$. – Jean-Claude Arbaut Dec 03 '18 at 00:01
  • @Jean-ClaudeArbaut I think I did that, does it look better? – Mike Dec 03 '18 at 00:07
  • @Jean-ClaudeArbaut Nice! I'll clarify that part. – Mike Dec 03 '18 at 00:12
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    See https://math.stackexchange.com/questions/1597326/degree-of-the-splitting-field-of-x3-5-over-mathbbq – Jean-Claude Arbaut Dec 03 '18 at 00:14

1 Answers1

2

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 $\mathrm e^{i\theta}$.
Bernard
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