Irreducible polynomial

**HTale** · February 14th 2009, 06:36 AM

I need to prove the following:

"Prove that $f(X)$ is irreducible in $\mathbb{Q}[X]$ if and only if $f(\alpha X + \beta)$ is irreducible in $\mathbb{Q}[X]$ "

I want to prove "if"; that is, $f(\alpha X + \beta)$ irreducible $\Rightarrow f(X)$ irreducible.

So, I think we have that $f(X)= a(X)b(X)$ , with one of $\deg(a), \deg(b) = \deg(f)$ .

The hint states that we should argue by contradiction. So, in this instance, we shall assume that $f(\alpha X + \beta)$ is reducible, and hence

$f(\alpha X + \beta) = g(\alpha X + \beta)h(\alpha X + \beta)$

where $\deg(g), \deg(h) < \deg(f)$

Now,

$f(\alpha X + \beta) \equiv [f(\alpha X + \beta) - f(X)] + f(X)$

and so,

$[f(\alpha X + \beta) - f(X)] + f(X) = g(\alpha X + \beta)h(\alpha X + \beta)$

we say, $f(\alpha X + \beta) - f(X)= \tilde{f}(\alpha X + \beta)$ , and so

$f(X) = g(\alpha X + \beta)h(\alpha X + \beta) - \tilde{f}(\alpha X + \beta)$

with $\deg(\tilde{f}) < \deg(g)$ , by the division algorithm. But, this is a contradiction, and so $\tilde{f}(\alpha X + \beta) = 0$ . Thus,

$f(X) = g(\alpha X + \beta)h(\alpha X + \beta) = a(X)b(X)$

which contradicts the fact that $\mathbb{Q}[X]$ is a unique factorisation domain.

I think this is wrong. I'd be really grateful if anyone could pick out the flaws in this proof, and if possible, provide a better (or, correct) proof.

Thanks in advance,

HTale.

**GaloisTheory1** · February 14th 2009, 07:30 AM

Originally Posted by HTale

I need to prove the following:

"Prove that $f(X)$ is irreducible in $\mathbb{Q}[X]$ if and only if $f(\alpha X + \beta)$ is irreducible in $\mathbb{Q}[X]$ "

I want to prove "only if"; that is, $f(\alpha X + \beta)$ irreducible $\Rightarrow f(X)$ irreducible.

minor error:

you are are trying to prove the "if" direction.

**HTale** · February 14th 2009, 08:58 AM

Originally Posted by GaloisTheory1

minor error:

you are are trying to prove the "if" direction.

Silly me! Error corrected; thanks for that. Are there any flaws in that proof, as far as you can see?

**ThePerfectHacker** · February 14th 2009, 10:23 AM

First of all $\alpha \not = 0$ ! Now define $\phi: \mathbb{Q}[X]\to \mathbb{Q}[X]$ by $\phi (f(X)) = f(\alpha X+\beta)$ . Since $\alpha\not = 0$ it means $\deg f = \deg \phi (f)$ . Say that $f(X) = f_1(X)f_2(X)$ where $\deg f_1,\deg f_2 < \deg f$ and $f_1,f_2\not \in \mathbb{Q}$ . This means, $\phi (f) = \phi (f_1)\phi (f_2)$ but $\deg \phi (f) = \deg (f)$ , $\deg \phi (f_1) = \deg f_1$ , $\deg \phi (f_2) = \deg f_2$ . It follows that $\deg \phi(f_1),\deg \phi(f_2) < \deg \phi (f)$ and $\phi(f_1),\phi(f_2) \not \in \mathbb{Q}$ , therefore $\phi (f) = f(\alpha X + \beta)$ is not irreducible. We have proved that if $f(X)$ is reducible then $f(\alpha X + \beta)$ is reducible, taking the contrapositive we have proven that if $f(\alpha X + \beta)$ is irreducible then $f(X)$ is irreducible. To prove the converse I am going to give you a hint

. Define $\theta : \mathbb{Q}[X] \to \mathbb{Q}[X]$ by $\theta (g(X)) = g\left( \tfrac{1}{\alpha}X - \tfrac{\beta}{\alpha}\right)$ (again $\alpha\not = 0$ ). Notice that $\phi (\theta(g(X))) = \theta (\phi (g(X))) = g(X)$ .

**HTale** · February 14th 2009, 10:58 AM

Originally Posted by ThePerfectHacker

First of all $\alpha \not = 0$ ! Now define $\phi: \mathbb{Q}[X]\to \mathbb{Q}[X]$ by $\phi (f(X)) = f(\alpha X+\beta)$ . Since $\alpha\not = 0$ it means $\deg f = \deg \phi (f)$ . Say that $f(X) = f_1(X)f_2(X)$ where $\deg f_1,\deg f_2 < \deg f$ and $f_1,f_2\not \in \mathbb{Q}$ . This means, $\phi (f) = \phi (f_1)\phi (f_2)$ but $\deg \phi (f) = \deg (f)$ , $\deg \phi (f_1) = \deg f_1$ , $\deg \phi (f_2) = \deg f_2$ . It follows that $\deg \phi(f_1),\deg \phi(f_2) < \deg \phi (f)$ and $\phi(f_1),\phi(f_2) \not \in \mathbb{Q}$ , therefore $\phi (f) = f(\alpha X + \beta)$ is not irreducible. We have proved that if $f(X)$ is reducible then $f(\alpha X + \beta)$ is reducible, taking the contrapositive we have proven that if $f(\alpha X + \beta)$ is irreducible then $f(X)$ is irreducible. To prove the converse I am going to give you a hint

. Define $\theta : \mathbb{Q}[X] \to \mathbb{Q}[X]$ by $\theta (g(X)) = g\left( \tfrac{1}{\alpha}X - \tfrac{\beta}{\alpha}\right)$ (again $\alpha\not = 0$ ). Notice that $\phi (\theta(g(X))) = \theta (\phi (g(X))) = g(X)$ .

Thanks a lot for your help! When you say $f_1,f_2\not \in \mathbb{Q}$ , are you saying that $f_1,f_2\not \in \mathbb{Q}[X]$ ? I'm not entirely sure what you are trying to say here.

**ThePerfectHacker** · February 14th 2009, 11:17 AM

Originally Posted by HTale

Thanks a lot for your help! When you say $f_1,f_2\not \in \mathbb{Q}$ , are you saying that $f_1,f_2\not \in \mathbb{Q}[X]$ ? I'm not entirely sure what you are trying to say here.

A polynomial $h(X)$ is in $\mathbb{Q}$ if and only if $h(X)$ is konstant.
That is all I meant.

**HTale** · February 14th 2009, 11:20 AM

Originally Posted by ThePerfectHacker

A polynomial $h(X)$ is in $\mathbb{Q}$ if and only if $h(X)$ is konstant.
That is all I meant.

Aha, in other words, non-unit polynomials. Thank you once again! I will try to attempt the second part of the proof, and post something on here soon

**ThePerfectHacker** · February 14th 2009, 11:37 AM

Originally Posted by HTale

Aha, in other words, non-unit polynomials. Thank you once again! I will try to attempt the second part of the proof, and post something on here soon

After you finish posting your proof for the problem you been assigned here is a good application of this theorem. Let $p$ be a prime number and consider the polynomial $X^{p-1} + X^{p-2} + ... + X + 1$ . This polynomial is always irreducible. See if you can prove this by $X\mapsto X+1$ (i.e. $\alpha = 1 \text{ and }\beta = 1$ ).

**HTale** · February 14th 2009, 03:09 PM

Originally Posted by ThePerfectHacker

After you finish posting your proof for the problem you been assigned here is a good application of this theorem. Let $p$ be a prime number and consider the polynomial $X^{p-1} + X^{p-2} + ... + X + 1$ . This polynomial is always irreducible. See if you can prove this by $X\mapsto X+1$ (i.e. $\alpha = 1 \text{ and }\beta = 1$ ).

I think in this instance, we have that $f(X) = \frac{X^p - 1}{X-1}$ , and so if we set $X\mapsto X+1$ , we get

$\frac{1}{X}((X+1)^p -1) = \sum^{p-1}_{r=0} {p \choose r}X^{p-r-1}$

and I think you apply Eisensteins criterion to finish the proof.

That's just a sketch of what I'd do. If it's correct, then I'll fill in the details and post the other proof and this to this thread.

HTale

**ThePerfectHacker** · February 14th 2009, 06:45 PM

Originally Posted by HTale

I think in this instance, we have that $f(X) = \frac{X^p - 1}{X-1}$ , and so if we set $X\mapsto X+1$ , we get

$\frac{1}{X}((X+1)^p -1) = \sum^{p-1}_{r=0} {p \choose r}X^{p-r-1}$

and I think you apply Eisensteins criterion to finish the proof.

That's just a sketch of what I'd do. If it's correct, then I'll fill in the details and post the other proof and this to this thread.

HTale

Yes

. The only issue I will take with your proof is that you wrote $f(X) = \frac{X^p - 1}{X-1}$ . This is not a geometric sequence, these are just polynomials (I mean we can add and multiply polynomials we cannot divide them). What I would do to be more formal is say that $f(X)(X-1) = X^p - 1$ - this can be easily seen by multiplying out the LHS. Now if you let $X\mapsto X+1$ then you get $Xf(X+1) = (X+1)^p - 1 = X\sum_{k=0}^{p-1}{{p}\choose {k+1}}X^k$ . We can "cancel" the $X$ (I say "cancel" because we are not really dividing since there is no such thing as division of polynomials, however, it is as if we are canceling) to get $f(X+1) = \sum_{k=1}^{p-1} {{p}\choose {k+1}}X^k$ . Now apply Eisenstein criterion by using the fact that $p$ divides ${p\choose {\ell}}$ if $1 < \ell < p$ .

**HTale** · February 15th 2009, 03:29 AM

One other question. If I set, say, $\alpha = 1$ , so that our polynomials are monic, could I conclude that this was the case for $\mathbb{Z}[X]$ ? I guess the maps you could express in a proof would be well defined.

**ThePerfectHacker** · February 15th 2009, 07:59 AM

Originally Posted by HTale

One other question. If I set, say, $\alpha = 1$ , so that our polynomials are monic, could I conclude that this was the case for $\mathbb{Z}[X]$ ? I guess the maps you could express in a proof would be well defined.

If the polynomials have integer coefficients then you can define $X\mapsto \pm X + \beta$ to get a transformed polynomial. This transformed polynomial is irreducible if and only if the original polynomial was irreducible by following the exact same proof except $\mathbb{Q}$ is replaced by $\mathbb{Z}$ . We need to avoid the case $|\alpha| > 1$ because we cannot define an inverse transform because we are in the integers.

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