What is the Galois group of 27 x 8 − 72 x 4 − 16 over the rationals?

Question

What is the Galois group of   27      x    8    −  72      x    4    −  16 over the rationals?

Eli Shaffer · Accepted Answer

The roots are easily found to be  ±                    4        (        3        +        2                  3                )            9        4    =  ±  a   (say)  ,    ±  i                    4        (        3        +        2                  3                )            9        4    ,    ±  (  1  ±  i  )                    2                  3                −        3            9        4    =  ±  (  1  ±  i  )  b   (say)with a,b being positive real numbers.The given polynomial is irreducible over       Q   and hence   [      Q    (  a  )  :      Q    ]  =  8. The splitting field L of the polynomial contains   a  ,  a  i and hence   i  ∈  L. But   i  ∉      Q    (  a  ) and hence   [      Q    (  a  ,  i  )  :      Q    ]  =  16It can be proved with some effort that   L  =      Q    (  a  ,  i  ) is the splitting field of the polynomial and hence the Galois group is of order 16.We can note that  a  b  =            12      4        3    =      c    3     (say)and next we show that   c  ∈  L. Since c is real we have in fact   c  ∈      Q    (  a  ). We can observe that      c    2    =  2      3    =            9              a        4            −      12        4    ∈      Q    (  a  )and                    9                  a          4                                    =        4                  c          2                +        12                                          =        4                  c          2                +                  c          4                                                  =                  c          2                (                  c          2                +        4        )                                          =                  c          2                (        4        +        2                  3                )                                          =                  c          2                (        1        +                  3                          )          2                    This implies  3      a    2    =  c  (  1  +      3    )ie  c  =            6              a        2                    2      +              c        2              ∈      Q    (  a  )It now follows that   b  =  c      /    (  3  a  )  ∈      Q    (  a  )  ⊂  L and L is the desired splitting field.The Galois group is       D          16       (dihedral group of order 16) as explained in another answer.

What is the Galois group of 27

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