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When considering the Galois group of the splitting field of the polynomial $x^3-2$, it is mentioned in my notes that $\sqrt[3]{2}$ can be mapped to $\sqrt[3]{2}$,$\sqrt[3]{2}\omega$ or $\sqrt[3]{2}\omega^2$, where $\omega$ is the cube root of unity. $\omega$ must be mapped to $\omega$ or $\omega^2$.

My question is why is this so? Sorry for the beginner question, but why can't $\sqrt[3]{2}$ be mapped to say $\omega$, or $\omega$ be mapped to say, 1?

Thank you very much for help.

  • 3
    Hint : Can you prove a number can only be mapped to one of the other roots in its minimal polynomial?2012-10-30
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    You can find an explanation [Example 7.2.14](http://www.maths.uq.edu.au/~victors/Algebra/7.pdf).2012-10-30
  • 2
    @Reader the link is dead, this seems like a very valuable post2015-10-18

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