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Application of cyclotomic polynomial

  1. Nov 30, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that [tex]{p \choose k} = \sum^{k+1}_{i=1} {p-i \choose p-k-1}[/tex] where [tex]\forall k < p \in \mathbb{Z}[/tex] and [tex]p[/tex] a prime.

    2. Relevant equations

    This is part (b) to a problem. Part (a) is showing that [tex]1 + x + x^2 + \cdots + x^{p-1}[/tex] is irreducible in [tex]\mathbb{Q}[x][/tex].

    3. The attempt at a solution

    I don't know how to apply part (a) to part (b)...Worked out the equality algebraically, then could probably do it by finite induction but think there has to be a shorter easier way that's supposed to follow from the previous part. Help.
  2. jcsd
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