Is polynomial 1 + x + x 2 + ⋯ + x p − 1 irreducible?

Question

Is polynomial   1  +  x  +      x    2    +  ⋯  +      x          p      −      1       irreducible?

Kristen Garrison · Accepted Answer

The polynomial can be rewritten as                     x        p            −      1              x      −      1      . Setting   x  =  (  y  +  1  ) has no effect on (ir)reducibility of the polynomial; doing so and applying the binomial theorem yields:            (      y      +      1              )        p            −      1              (      y      +      1      )      −      1        =      1    y        ∑          k      =      1              p                          (                    p      k                      )                  y          k        =      y          p      −      1        +                    (                    p              p        −        1                            )                  y          p      −      2        +  …  +                    (                    p      2                      )                  y    1    +                    (                    p      1                      )                  y    0  Since all the coefficients apart from the leading one are divisible by p and the constant term is too small to be divisible by       p    2  , we can apply the criterion and conclude that the polynomial is irreducible over rationals (and thus also over integers).

Is polynomial 1+x+x^2+⋯+x^(p−1) irreducible?

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