(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider [tex] f(x) = x^3-5 [/tex]

and its splitting field [tex] K = Q(5^{1/3}, \omega) [/tex]

where [tex] \omega = e^{2 \pi i/3} [/tex]

Show that [tex] B = \{1, 5^{1/3}, 5^{2/3}, \omega, \omega 5^{1/3} , \omega 5^{2/3} \} [/tex]

is a vector space basis for K over Q.

3. The attempt at a solution

I am just a bit confused. Since [tex] 5^{1/3} [/tex] and [tex] \omega [/tex] are in K, and K is a field, then [tex] B'= \{ \omega ^2, \omega ^2 5^{1/3}, \omega ^2 5^{2/3} \} \subseteq K [/tex]

But how can we get any of these elements using only the shown basis B with scalars in Q? I would think that B+B' would be the vector space basis.

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# Splitting Field Proof

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