# Prove Polynomial is Irreducible

## Homework Statement

Show the polynomial
$$f(x) = x^p + x^{p-1} + ... + x - 1$$
is irreducible over $Z_p$ where p is a prime.

## 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:

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

So, showing this polynomial is irreducible over $Z_p$ is equivalent to the original problem. But again, I'm stuck here :).

BTW, no real need for the reciprocal as I just realized that you can derive $f(x+1) = x^p + x^{p-1} - 1$ similarly.

In case anyone is interested in the solution:

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