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How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?

Right now I'm able to prove that it has no roots and that it is separable, but I have not a clue as to how to prove it is irreducible. Any ideas?

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When reading recently an article about the Artin-Schreier theorem, some properties of the so-called Artin extensions were used, and, if no mistakes occur here, those are intimately related to the polynomials of the form $x^p-x+a$. Is there indeed any error that occur? and is there any reference to know more in this direction? Thanks in advance. –  awllower Nov 13 '11 at 14:05
    
@awllower: This question may get you started? –  Jyrki Lahtonen Nov 13 '11 at 16:39
    
@JyrkiLahtonen: Thanks for the question. I really appreciate this. –  awllower Nov 13 '11 at 16:46
    
BTW: Closing this question will not affect the reputation Ayman has earned. My understanding is that he can even receive more upvotes after closure. If the question were deleted that would make him lose the points, but closure is different. –  Jyrki Lahtonen Aug 3 '13 at 10:25
    
If that is the case, then it is fine :) –  Praphulla Koushik Aug 3 '13 at 10:29

5 Answers 5

up vote 19 down vote accepted

Greg Martin and zyx have given you IMHO very good answers, but they rely on a few basic facts from Galois theory and/or group actions. Here is a more elementary but also longer approach.

Because we are in char $p$, the polynomial $g(x)=x^p-x$ has the property $$g(x_1+x_2)=g(x_1)+g(x_2).$$ By little Fermat we know that $g(k)=k^p-k=0$ for all $k\in F_p$. Therefore, if $r$ is one of the roots of $f(x)=x^p-x+a$, then $f(r+k)=g(r+k)+a=f(r)+g(k)=0$, so all the elements $r+k, k=1,2,\ldots,p-1$ are roots of $f(x)$, and as there are $p$ of them, they must be all the roots. It sounds like you have already shown that $r$ cannot be an element of $F_p$.

Now assume that $f(x)=f_1(x)f_2(x)$, where both factors $f_1(x),f_2(x)\in F_p[x]$. From the above consideration we can deduce that $$ f_1(x)=\prod_{k\in S}(x-(r+k)), $$ where $S$ is some subset of the field $F_p$. Write $\ell=|S|=\deg f_1(x)$. Expanding the product we see that $$ f_1(x)=x^\ell-x^{\ell-1}\sum_{k\in S}(r+k)+\text{lower degree terms}. $$ This polynomial was assumed to have coefficients in the field $F_p$. From the above expansion we read that the coefficient of degree $\ell-1$ is $|S|\cdot r+\sum_{k\in S}k$. This is an element of $F_p$, if and only if the term $|S|\cdot r\in F_p$. Because $r\notin F_p$, this can only happen, if $|S|\cdot1_{F_p}=0_{F_p}$. In other words $f_1(x)$ must be either of degree zero or of degree $p$.

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Well done, it is a good proof. –  awllower Nov 13 '11 at 14:01
    
I love your proof. One thing is bothering me, though. And It's probably really obvious. How do you know that if $|S|\cdot r\in F_p$, then $|S|$ must be a multiple of $p$ in order for $|S|\cdot r$ to be in $F_p$? I know it has to do with the fact that $r\notin F_p$, and I have some intuition for it, but I don't know how to prove it. –  Alison Nov 14 '11 at 4:21
    
@MathMastersStudent: If $|S|$ is not a multiple of $p$, then $|S|\cdot 1$ is an invertible element of $F_p$. So if $|S|\cdot r= b$ with $b\in F_p$, then $r=b|S|^{-1}$ would be in the prime field $F_p$ as well contradicting known facts. –  Jyrki Lahtonen Nov 14 '11 at 7:20
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Great answer as usual, Jyrki: +1. Just to show you how pathologically nit-picking some guys are, I would write $|S|\cdot 1_{F_p}=0_{F_p}$ rather than $|S|=0_{F_p}$, since $S$ is an integer and the integers are not included in a finite field... –  Georges Elencwajg Jul 24 '13 at 8:05

$x \to x^p$ is an automorphism sending $r$ to $r-a$ for any root $r$ of the polynomial. This operation is cyclic of order $p$, so that one can get from any root to any other by applying the automorphism several times. The Galois group thus acts transitively on the roots, which is equivalent to irreducibility.

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Impressively elegant, zyx: +1 –  Georges Elencwajg Jul 24 '13 at 8:12

I think the following idea works. Let $f(x) = x^p-x+a$. They key observation is that $f(x+1)=f(x)$ in the field of $p$ elements. Now factor $f(x) = g_1(x) \cdots g_k(x)$ as a product of irreducibles. Sending $x$ to $x+1$ must therefore permute the factors $\{ g_1(x), \dots, g_k(x) \}$. But sending $x$ to $x+1$ $p$ times in a row comes back to the original polynomial, so this permutation of the $k$ factors has order dividing $p$. It follows that either every $g_j(x)$ is fixed by sending $x$ to $x+1$ - which I think is a property that no nonconstant polynomial of degree less than $p$ can have, but that needs proof - or else there are $k=p$ factors, which can only happen in the case $a=0$.

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$f(x)$ is separable since its derivative is $f'(x) = -1 \ne 0$.

Suppose $\theta$ is a root of $f(x) = x^p - x + a$. Using the Frobenius automorphism, we have: \begin{align} f(\theta + 1) &= (\theta + 1)^p - (\theta + 1) - a\\ &= \theta^p + 1^p - \theta - 1 + a\\ &= \theta^p - \theta + a\\ &= f(\theta) = 0 \end{align}

Thus, by induction, if $\theta$ is a root of $f(x)$, then $\theta + j$ is also a root for all $j \in \mathbb F_p$.

By above, if $f(x)$ were to have a root in $\mathbb F_p$, then $0$ would a be a root too, but this contradicts $a \ne 0$. Thus, $f(x)$ has no roots in $\mathbb F_p$. (This can also be shown using Fermat's little theorem.)

Suppose $\theta$ is a root of $f(x)$ in some extension of $\mathbb F_p$. We know that $\theta + j$ is also a root for all $j \in \mathbb F_p$. Since $f(x)$ is of degree $p$, these are all of the roots of $f(x)$.

Clearly, $\mathbb F_p(\theta) = \mathbb F_p(\theta + j)$ for all $j \in \mathbb F_p$. Thus, all $\{\theta + j\}$ have the same degree over $\mathbb F_p$. Since $f(x)$ is separable, it follows that $f(x)$ must be the product of all minimal polynomials of $\{\theta + j\}$. Suppose the minimal polynomials have degree $m$. We have $p = km$ for some $k$. Since $p$ is prime, either $m = 1$; hence $\theta \in \mathbb F_p$, a contradiction. Or $k = 1$; hence $f(x)$ is irreducible because it's the minimal polynomial.

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Ah, I already accepted it. I want to accept it another time but it will get as unaccepted :P Thank You for this wonderful proof :) :) :) –  Praphulla Koushik Aug 3 '13 at 10:03
    
A good one! ${}$ –  Jyrki Lahtonen Aug 3 '13 at 10:06
    
Happy to help! I have it in my notes. I don't actually remember if I came up with it myself or found it somewhere. –  Ayman Hourieh Aug 3 '13 at 10:11
    
What ever may be the source, Your Answer is good :) –  Praphulla Koushik Aug 3 '13 at 10:16

$x^p-x+a$ divides $x^{p^p}-x$. If $f$ is an irreducible divisor of $x^p-x+a$ of degree $d$ then $\mathbf{Z}_p[x]/f$ will be a subfield of the field with $p^p$ elements so $p^p = (p^d)^e$ and so $d=1$ or $e=1$. since $x^p-x+a$ has no roots $e=p.$

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