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I could not find any proof on the Internet. I am looking for a formal proof with an explanation for the uninitiated (my knowledge of Galois theory is very basic). With geometrically constructible I mean with compass and straightedge.

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Some cube roots are constructible. Could you be a bit more specific? Does this overview help you at all? –  Jyrki Lahtonen Jul 2 '12 at 10:49
Well, this has less to do with Galois theory and more to do with constructibility. The first step is to show that a length is constructible if and only of it can be expressed using addition, subtraction, multiplication, division, and square roots... –  Zhen Lin Jul 2 '12 at 10:52

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Since the minimal algebraic polynomial for the $2^{1/3}$ is $x^3-2=0$ and for $2^{1/3}$ to be constructible , the degree of this polynomial needs to be a power of $2$, but since $3$ is not a power of $2$, hence $2^{1/3}$ is not constructible. Read this http://www2.math.uu.se/~svante/papers/sjN8.pdf

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