<prompt> Nambari changamano alpha_1, alpha_2, alpha_3, na alpha_4 ni mizizi minne tofauti ya mlinganyo x42x320. Amua seti isiyopangwa alpha_1alpha_2 alpha_3alpha_4, alpha_1alpha_3 alpha_2alpha_4, alpha_1alpha_4 alpha_2alpha_3.  <response>: Kuajiri polimanomia za msingi za ulinganifu s_1 alpha_1alpha_2alpha_3alpha_4 -2, s_2 alpha_1alpha_2 alpha_1alpha_3 alpha_1alpha_4 alpha_2alpha_2 alpha_2 alpha_3 alpha_3alpha_4 0, s_3 alpha_1alpha_2alpha_3 alpha_2alpha_3alpha_4 alpha_3alpha_4alpha_1 alpha_4alpha_1alpha_2 0, na s_4alpha_ alpha_3alpha_4 2 tunazingatia polinomia Px x-alpha_1alpha_2alpha_3alpha_4x-alpha_1alpha_3alpha_2alpha_4x-alpha_1alpha_4alpha_2alpha_3Kwa sababu P ina ulinganifu kuhusiana na alpha_1, alpha_2, alpha_3, alpha_4, tunaweza kueleza viambajengo vya umbo lake lililopanuliwa kulingana na polynomia za kimsingi linganifu. Tunahesabu kuanzaeqnarrayPx x3 - s_2x2 s_3s_1-4s_4x -s_32-s_4s_12s_4s_2 x3 - 8x - 8 x2x2-2x-4endeqnarrayMizizi ya Px ni -2 na 1 pm sqrt5, kwa hivyo jibu ni 1pmsqrt5,-2 .textbfRemarks: Ni rahisi kupata viambajengo vya x2 na x kwa upanuzi, na istilahi isiyobadilika inaweza kukokotwa bila upanuzi kamili na mtengano wa alpha_1alpha_2alpha_3. alpha_4alpha_1alpha_3alpha_2alpha_4alpha_1alpha_4alpha_2alpha_3 kwa kubainisha kuwa maneno ya nonzero ya shahada ya 6 pekee katika s_1, s_2, s_3, na s_4 ni s_1 6 na s_4s_12. Polynomial ya jumla P iliyojengwa hapa inaitwa cubic solvent na inatokana na nadharia ya Galois. 