Prove Polynomial is Irreducible

  • #1

Homework Statement


Show the polynomial
[tex]
f(x) = x^p + x^{p-1} + ... + x - 1
[/tex]
is irreducible over [itex]Z_p[/itex] where p is a prime.


Homework Equations





The Attempt at a Solution


I know f(x) has no roots in Zp, but other than that, I'm stuck. Thanks for the help.
 
Last edited:

Answers and Replies

  • #2
I tried:
Let [itex]g(x) = x^{p}f(1/x)[/itex] be the reciprocal of f(x). g(x) is irreducible iff f(x) is irreducible. We have
[tex]g(x) = -x^p + x^{p-1} + ... + x + 1[/tex]
[tex]-g(x) = x^p - x^{p-1} - ... - x - 1[/tex]
[tex]-(x - 1)g(x) = x^{p+1} - 2x^p + 1[/tex]
[tex]-xg(x+1) = (x+1)^{p+1} - 2(x+1)^p + 1 = x^{p+1} - x^p + x[/tex]
[tex]-g(x+1) = x^p - x^{p-1} + 1[/tex]

So, showing this polynomial is irreducible over [itex]Z_p[/itex] is equivalent to the original problem. But again, I'm stuck here :).
 
  • #3
BTW, no real need for the reciprocal as I just realized that you can derive [itex] f(x+1) = x^p + x^{p-1} - 1[/itex] similarly.
 
  • #4
In case anyone is interested in the solution:

The reciprocal of f(x+1) is [itex]g(x) = 1 + x - x^p[/itex]. But -g(x) is an irreducible trinomial. It follows f(x) is irreducible.
 

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