Let F = C ( X , Y , Z ) be rational function field.And let K be the splitting field of F ( T ) = T 6 − X T 4 + Y T 2 − Z.Please Find all galois extensions L / F such that L ⊂ K and [ L : F ] = 6.I'm stuck on this problem.

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Let   F  =      C    (  X  ,  Y  ,  Z  ) be rational function field.And let   K be the splitting field of   F  (  T  )  =      T    6    −  X      T    4    +  Y      T    2    −  Z.Please Find all galois extensions   L      /    F such that   L  ⊂  K and   [  L  :  F  ]  =  6.I&#039;m stuck on this problem.

Gallichi5mtwt · Accepted Answer

First, we find the degree of the splitting field of   P  (  T  )  =      T    6    −  X      T    4    +  Y      T    2    −  Z. We note that the discriminant   D of   G  (  T  )  =      T    3    −  X      T    2    +  Y  T  −  Z is not a perfect square: indeed, the discriminant is a polynomial of degree 3 in   X, thus can&#039;t be a perfect square. Let the roots of   G be       X    1    ,      X    2    ,      X    3  , then the splitting field of   G over   F is   M  =  F  (      X    1    ,      X    2    ,      X    3    )  =  F  (      X    1    ,      D    ), and has degree 6 over   F. Then, we see that  K  =  M  (            X      1        ,            X      2        ,            X      3        )Fortunately for us, we actually don&#039;t need to know the exact degree of this extension - it suffices to know that it divides 8, so that the degree   [  K  :  F  ] divides 48. (The exact degree is, in fact, 48.) Why is this helpful? Well, if   L      /    F is Galois, then any irreducible polynomial in   F  [  X  ] splits into factors of equal degree over   L. In particular,   G  (  T  ) either splits completely or remains irreducible. If the former happens, then by degree considerations we have   L  =  M. Otherwise,   G remains irreducible over   L; which implies that the degree   [  K  :  L  ] is divisible by 3. But the degree   [  L  :  F  ] is already divisible by 3; so we obtain the absurd result that   [  K  :  F  ], and thus 48, is divisible by 9. Thus,   M is the unique intermediate Galois extension of degree 6 over   F.

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