Let f ( x ) = x 3 − 3 x 2 + 3 x + 1 , g ( x ) = x 3 + 3 x 2 + 3 x + 3. I would like to show that Q [ x ] / ⟨ f ( x ) ⟩ ≅ Q [ x ] / ⟨ g ( x ) ⟩

Question

Let   f  (  x  )  =      x    3    −  3      x    2    +  3  x  +  1  ,  g  (  x  )  =      x    3    +  3      x    2    +  3  x  +  3. I would like to show that       Q    [  x  ]      /        ⟨    f    (    x    )    ⟩    ≅      Q    [  x  ]      /        ⟨    g    (    x    )    ⟩

Korotnokby · Accepted Answer

First, for any   h  ∈      Q    [  x  ], we can define       Φ    h    :      Q    [  x  ]  →      Q    [  x  ] by       Φ    h    (  F  )  =  F  ∘  h (where   ∘ denotes compositon). You can check directly that       Φ    h   is a homomorphism for all h. Moreover,       Φ    x    =      id                  Q            [      x      ]      , and       Φ                  h        1            ∘              h        2              =      Φ                  h        2              ∘      Φ                  h        1             for all       h    1    ,      h    2    ∈      Q    [  x  ]Now notice that   (  x  −  2  )  ∘  (  x  +  2  )  =  (  x  +  2  )  ∘  (  x  −  2  )  =  x, so       Φ          x      −      2        ∘      Φ          x      +      2        =      id                  Q            [      x      ]        =      Φ          x      +      2        ∘      Φ          x      −      2      . Thus,       Φ          x      +      2       is an automorphism of       Q    [  x  ] with inverse       Φ          x      −      2      . Note that       Φ          x      +      2        (  f  )  =  g in your example.In general, if   α  :  R  →  S is a ring isomorphism and I is a two-sided ideal of R, then   α  (  I  ) is a two-sided ideal of S, and   α induces a ring isomorphism   R      /    I  →  S      /    α  (  I  ), given by   r  +  I  ↦  α  (  r  )  +  α  (  I  )So, we conclude that       Φ          x      +      2       induces an isomorphism      Q    [  x  ]      /    ⟨  f  ⟩  →      Q    [  x  ]      /        Φ          x      +      2        (  ⟨  f  ⟩  )  =      Q    [  x  ]      /    ⟨  g  ⟩  .

Let f ( x ) = x 3 </msup> − 3 x 2 </msup>

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