Fields1

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Let \alpha ={\sqrt[ {3}]{5}} and \zeta _{3}={\frac  {1}{2}}(-1+{\sqrt  {-3}}) and note that the roots of f\, in {\mathbb  {C}} are \alpha ,\,\alpha \zeta _{3},\,\alpha \zeta _{3}^{2}. Then the splitting field of f(x)=x^{3}-5\, over {\mathbb  {Q}} is E={\mathbb  {Q}}(\alpha ,\alpha \zeta _{3},\alpha \zeta _{3}^{2})={\mathbb  {Q}}(\alpha ,\alpha \zeta _{3})={\mathbb  {Q}}(\alpha ,\zeta _{3}).

Note the minimal polynomial of \alpha ={\sqrt[ {3}]{5}} over {\mathbb  {Q}} is m_{{\alpha ,{\mathbb  {Q}}}}(x)=x^{3}-5\,, so [{\mathbb  {Q}}(\alpha ):{\mathbb  {Q}}]=3. Further, the minimal polynomial of \zeta _{3}\, over {\mathbb  {Q}}(\alpha ) is m_{{\zeta _{3},{\mathbb  {Q}}(\alpha )}}(x)=x^{2}+x+1, so [{\mathbb  {Q}}(\alpha )(\zeta _{3}):{\mathbb  {Q}}(\alpha )]=2. Thus, [E:{\mathbb  {Q}}]=[{\mathbb  {Q}}(\alpha ,\zeta _{3}):{\mathbb  {Q}}(\alpha )]\cdot [{\mathbb  {Q}}(\alpha ):{\mathbb  {Q}}]=2\cdot 3=6.


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