Let C ( x , y ) be a degree 2 extension of C ( x ) where y is a root of p ( Z ) = Z 2 + ( x 2 + 1 ).Is it true that C ( x , y ) is not a rational function field? In other words, there is no z ∈ C ( x , y ) such that C ( x , y ) = C ( z )

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Let       C    (  x  ,  y  ) be a degree 2 extension of       C    (  x  ) where   y is a root of   p  (  Z  )  =      Z    2    +  (      x    2    +  1  ).Is it true that       C    (  x  ,  y  ) is not a rational function field? In other words, there is no   z  ∈      C    (  x  ,  y  ) such that       C    (  x  ,  y  )  =      C    (  z  )

reflam2kfnr · Accepted Answer

GeometricallyThe field       C    (  x  ,  y  ) is a rational function field because it is the field of rational functions of a smooth affine conic, namely the conic   C  ⊂            A              C        2   with equation       y    2    +      x    2    +  1  =  0.AlgebraicallyMore explicitly       C    (  x  ,  y  )  =      C    (  t  ) with   t  =            y      −      i        x  :Writing       C    (  x  ,  y  )  =                    C            (      x      )      [      Y      ]              (              Y        2            +              x        2            +      1      )       an explicit isomorphism is given by  f  :      C    (  x  ,  y  )                    →                    ≅                  C    (  t  )where   f  (  x  )  =            −      2      i      t              1      +              t        2             and   f  (  y  )  =  i            1      −              t        2                    1      +              t        2            .The inverse isomorphism is determined by       f          −      1        (  t  )  =            y      −      i        x  .The link between both approachesWe fix the point   P  =  (  0  ,  i  )  ∈  C and consider the line   y  =  i  +  t  x through P with slope t (notice that then   t  =            y      −      i        x   as above ).That line cuts the conic C in P and in the point       P    t    =  (            −      2      i      t              1      +              t        2              ,  i            1      −              t        2                    1      +              t        2              ) and this explains the formulas brutally parachuted in the previous section.

Let <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C </mrow> ( x

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