Let f = X 2 + 1 ∈ F 5 [ X ] and α = X + ⟨ f ⟩ ∈ R. Show that α ∈ R ∗ and that | α | = 4 in R ∗

Question

Let   f  =      X    2    +  1  ∈            F        5    [  X  ] and   α  =  X  +  ⟨  f  ⟩  ∈  R. Show that   α  ∈      R    ∗   and that   |  α  |  =  4 in       R    ∗

Clare Chung · Accepted Answer

Working in R means working with polynomials in             F        5    [  X  ] modulo the polynomial   f  =      X          2        +  1So you have       α          2        =      X          2        ≡  −  1    (  mod    f  ), and       α          4        ≡  (  −  1      )          2        =  1    (  mod    f  ). This shows that   α is invertible, with inverse       α          3        ≡  −  X    (  mod    f  ). (This is because   1  ≡      α          4        ≡  α      α          3          (  mod    f  ).Also, since       α          4        ≡  1    (  mod    f  ), the order of   α is a divisor of 4. But since       α          2        ≡  −  1  ≢  1    (  mod    f  ), the order does not divide 2, and thus it is indeed 4

Let f=X^2+1in FF_5[X], R=FF_5[X]/⟨f⟩ and alpha=X+⟨f⟩ in RR. Show that alpha in RR^∗ and that |alpha|=4 in RR^∗.

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