- #1

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the polynomial x^4+8x+12=0 has the Galois group A4. I have all its roots, but can't figure out its splitting field. The roots are

[tex]\alpha_1=\sqrt{2}(\sqrt{\cos{(\pi/9)}}+i\sqrt{\cos{(2\pi/9)}}+i\sqrt{\cos{(4\pi/9)}})[/tex]

[tex]\alpha_2=\sqrt{2}(\sqrt{\cos{(\pi/9)}} - i\sqrt{\cos{(2\pi/9)}}-i\sqrt{\cos{(4\pi/9)}})[/tex]

[tex]\alpha_3=\sqrt{2}(-\sqrt{\cos{(\pi/9)}} + i\sqrt{\cos{(2\pi/9)}}-i\sqrt{\cos{(4\pi/9)}})[/tex]

[tex]\alpha_4=\sqrt{2}(-\sqrt{\cos{(\pi/9)}} - i\sqrt{\cos{(2\pi/9)}}+i\sqrt{\cos{(4\pi/9)}})[/tex]

[tex]\alpha_1=\sqrt{2}(\sqrt{\cos{(\pi/9)}}+i\sqrt{\cos{(2\pi/9)}}+i\sqrt{\cos{(4\pi/9)}})[/tex]

[tex]\alpha_2=\sqrt{2}(\sqrt{\cos{(\pi/9)}} - i\sqrt{\cos{(2\pi/9)}}-i\sqrt{\cos{(4\pi/9)}})[/tex]

[tex]\alpha_3=\sqrt{2}(-\sqrt{\cos{(\pi/9)}} + i\sqrt{\cos{(2\pi/9)}}-i\sqrt{\cos{(4\pi/9)}})[/tex]

[tex]\alpha_4=\sqrt{2}(-\sqrt{\cos{(\pi/9)}} - i\sqrt{\cos{(2\pi/9)}}+i\sqrt{\cos{(4\pi/9)}})[/tex]

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