What is the minimal polynomial of α = 3 1 / 2 1 + 2 1 / 3 over Q ?

Question

What is the minimal polynomial of   α  =            3              1                  /                2                    1      +              2                  1                      /                    3                     over       Q  ?

Mathias Patrick · Accepted Answer

Since,   α  =            3              1                  /                2                    1      +              2                  1                      /                    3                      =      3          −      1              /            2        (  1  −      2          1              /            3        +      2          2              /            3        ), let   β  =      α    2    =      2          2              /            3        −  1. Then      [                                        β            0                                                            β            1                                                            β            2                                                            β            3                                ]    =      [                            1                          0                          0                                      −          1                          0                          1                                      1                          2                          −          2                                      3                          −          6                          3                      ]        [                            1                                                  2                          1                              /                            3                                                                        2                          2                              /                            3                                            ]  With a bit of linear algebra, or perhaps inspection, it is not too difficult to see that      β    3    +  3      β    2    +  3  β  −  3  =  0Therefore,      α    6    +  3      α    4    +  3      α    2    −  3  =  0which is irreducible by Eisenstein&#039;s Criterion.

What is the minimal polynomial of α = 3 <mrow class=

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