Galois

May 2009
15
0
Hi
Let \alpha{=2^1/3}, \omega {=e^2\pi {i/3)}}

how can we show Q(\alpha ,\omega } is a the splitting field of t^3-2

Thank you
 
May 2010
95
38
Hi
Let \alpha{=2^1/3}, \omega {=e^2\pi {i/3)}}

how can we show Q(\alpha ,\omega } is a the splitting field of t^3-2

Thank you
Just check t^3-2 \in Q[t] splits in Q(\alpha ,\omega) and verify that all the roots of t^3-2 \in Q[t] belong to Q(\alpha ,\omega). Make sure that t^3-2 \in Q[t] does not factor completely into linear factors in any proper subfield of Q(\alpha ,\omega) containing Q.