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Prove that the polynomial $x^5+2x+1$ is irreducible over $mathbbZ$.
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Show that there do not exist polynomials $p(x)$ and $q(x)$ , each having integer coefficients and of degree greater than or equal to 1 such that: $$p(x)cdot q(x) = x^5 + 2x + 1 $$
My approach
I found that only possible integral root of the $x^5 + 2x + 1 = 0 $ is ± 1 and these are not the zeroes of the polynomial. Hence obviously it doesn't have any linear factor. So i assumed $q(x)$ to be a third degree polynomial and $p(x)$ to be a second degree polynomial . That is ;$$p(x)=x^2+ax+b$$ $$q(x)=x^3+cx^2+dx+e$$ But i don't know how to solve it further. So please help me with this. And if there is some other approach to this question, then please share it .
abstract-algebra polynomials irreducible-polynomials
edited yesterday
Batominovski
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asked yesterday
Identicon
808
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up vote
3
down vote
favorite
Show that there do not exist polynomials $p(x)$ and $q(x)$ , each having integer coefficients and of degree greater than or equal to 1 such that: $$p(x)cdot q(x) = x^5 + 2x + 1 $$
My approach
I found that only possible integral root of the $x^5 + 2x + 1 = 0 $ is ± 1 and these are not the zeroes of the polynomial. Hence obviously it doesn't have any linear factor. So i assumed $q(x)$ to be a third degree polynomial and $p(x)$ to be a second degree polynomial . That is ;$$p(x)=x^2+ax+b$$ $$q(x)=x^3+cx^2+dx+e$$ But i don't know how to solve it further. So please help me with this. And if there is some other approach to this question, then please share it .
abstract-algebra polynomials irreducible-polynomials
edited yesterday
Batominovski
22.3k22676
asked yesterday
Identicon
808
5
Often it is good to check $pmod p$ for small primes $p$, just because the algebra is so much easier. In this case, the polynomial is irreducible $pmod 3$.
– lulu
yesterday
1
$c+a=0$ because no $x^4.$ Substitute back $c = -a$ and start again. What is the coefficient of $x^3 ; ?$
– Will Jagy
yesterday
1
Two choices, either $b=e=1$ or $b=e=-1.$ There is enough information to finish both cases using $c=-a$ and finding contradictions for each, after finding $d ; ; .$
– Will Jagy
yesterday
1
+1 to all. But I think we should compose a thread covering a compendium of techniques. This types of questions, while not as numerous as those of $a^bmod n$ or integration by partial fraction or..., are still frequent enough that it may serve a point to prepare an abstract duplicate.
– Jyrki Lahtonen
yesterday
@JyrkiLahtonen Like the idea, there are some questions that somehow attempted that, as Methods to see if a polynomial is irreducible. Although I wonder how much this can be treated as a duplicate since specific polynomial might require different method to be solved and it might not be apparent to which method does the duplicate link refer to...
– Sil
yesterday
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Show that there do not exist polynomials $p(x)$ and $q(x)$ , each having integer coefficients and of degree greater than or equal to 1 such that: $$p(x)cdot q(x) = x^5 + 2x + 1 $$
My approach
I found that only possible integral root of the $x^5 + 2x + 1 = 0 $ is ± 1 and these are not the zeroes of the polynomial. Hence obviously it doesn't have any linear factor. So i assumed $q(x)$ to be a third degree polynomial and $p(x)$ to be a second degree polynomial . That is ;$$p(x)=x^2+ax+b$$ $$q(x)=x^3+cx^2+dx+e$$ But i don't know how to solve it further. So please help me with this. And if there is some other approach to this question, then please share it .
abstract-algebra polynomials irreducible-polynomials
edited yesterday
Batominovski
22.3k22676
asked yesterday
Identicon
808
Show that there do not exist polynomials $p(x)$ and $q(x)$ , each having integer coefficients and of degree greater than or equal to 1 such that: $$p(x)cdot q(x) = x^5 + 2x + 1 $$
My approach
I found that only possible integral root of the $x^5 + 2x + 1 = 0 $ is ± 1 and these are not the zeroes of the polynomial. Hence obviously it doesn't have any linear factor. So i assumed $q(x)$ to be a third degree polynomial and $p(x)$ to be a second degree polynomial . That is ;$$p(x)=x^2+ax+b$$ $$q(x)=x^3+cx^2+dx+e$$ But i don't know how to solve it further. So please help me with this. And if there is some other approach to this question, then please share it .
abstract-algebra polynomials irreducible-polynomials
edited yesterday
Batominovski
22.3k22676
asked yesterday
Identicon
808
edited yesterday
Batominovski
22.3k22676
edited yesterday
Batominovski
22.3k22676
edited yesterday
Batominovski
22.3k22676
22.3k22676
asked yesterday
Identicon
808
asked yesterday
Identicon
808
asked yesterday
Identicon
808
808
5
Often it is good to check $pmod p$ for small primes $p$, just because the algebra is so much easier. In this case, the polynomial is irreducible $pmod 3$.
– lulu
yesterday
1
$c+a=0$ because no $x^4.$ Substitute back $c = -a$ and start again. What is the coefficient of $x^3 ; ?$
– Will Jagy
yesterday
1
Two choices, either $b=e=1$ or $b=e=-1.$ There is enough information to finish both cases using $c=-a$ and finding contradictions for each, after finding $d ; ; .$
– Will Jagy
yesterday
1
+1 to all. But I think we should compose a thread covering a compendium of techniques. This types of questions, while not as numerous as those of $a^bmod n$ or integration by partial fraction or..., are still frequent enough that it may serve a point to prepare an abstract duplicate.
– Jyrki Lahtonen
yesterday
@JyrkiLahtonen Like the idea, there are some questions that somehow attempted that, as Methods to see if a polynomial is irreducible. Although I wonder how much this can be treated as a duplicate since specific polynomial might require different method to be solved and it might not be apparent to which method does the duplicate link refer to...
– Sil
yesterday
add a comment |Â
5
Often it is good to check $pmod p$ for small primes $p$, just because the algebra is so much easier. In this case, the polynomial is irreducible $pmod 3$.
– lulu
yesterday
1
$c+a=0$ because no $x^4.$ Substitute back $c = -a$ and start again. What is the coefficient of $x^3 ; ?$
– Will Jagy
yesterday
1
Two choices, either $b=e=1$ or $b=e=-1.$ There is enough information to finish both cases using $c=-a$ and finding contradictions for each, after finding $d ; ; .$
– Will Jagy
yesterday
1
+1 to all. But I think we should compose a thread covering a compendium of techniques. This types of questions, while not as numerous as those of $a^bmod n$ or integration by partial fraction or..., are still frequent enough that it may serve a point to prepare an abstract duplicate.
– Jyrki Lahtonen
yesterday
@JyrkiLahtonen Like the idea, there are some questions that somehow attempted that, as Methods to see if a polynomial is irreducible. Although I wonder how much this can be treated as a duplicate since specific polynomial might require different method to be solved and it might not be apparent to which method does the duplicate link refer to...
– Sil
yesterday
5
5
Often it is good to check $pmod p$ for small primes $p$, just because the algebra is so much easier. In this case, the polynomial is irreducible $pmod 3$.
– lulu
yesterday
Often it is good to check $pmod p$ for small primes $p$, just because the algebra is so much easier. In this case, the polynomial is irreducible $pmod 3$.
– lulu
yesterday
1
1
$c+a=0$ because no $x^4.$ Substitute back $c = -a$ and start again. What is the coefficient of $x^3 ; ?$
– Will Jagy
yesterday
$c+a=0$ because no $x^4.$ Substitute back $c = -a$ and start again. What is the coefficient of $x^3 ; ?$
– Will Jagy
yesterday
1
1
Two choices, either $b=e=1$ or $b=e=-1.$ There is enough information to finish both cases using $c=-a$ and finding contradictions for each, after finding $d ; ; .$
– Will Jagy
yesterday
Two choices, either $b=e=1$ or $b=e=-1.$ There is enough information to finish both cases using $c=-a$ and finding contradictions for each, after finding $d ; ; .$
– Will Jagy
yesterday
1
1
+1 to all. But I think we should compose a thread covering a compendium of techniques. This types of questions, while not as numerous as those of $a^bmod n$ or integration by partial fraction or..., are still frequent enough that it may serve a point to prepare an abstract duplicate.
– Jyrki Lahtonen
yesterday
+1 to all. But I think we should compose a thread covering a compendium of techniques. This types of questions, while not as numerous as those of $a^bmod n$ or integration by partial fraction or..., are still frequent enough that it may serve a point to prepare an abstract duplicate.
– Jyrki Lahtonen
yesterday
@JyrkiLahtonen Like the idea, there are some questions that somehow attempted that, as Methods to see if a polynomial is irreducible. Although I wonder how much this can be treated as a duplicate since specific polynomial might require different method to be solved and it might not be apparent to which method does the duplicate link refer to...
– Sil
yesterday
@JyrkiLahtonen Like the idea, there are some questions that somehow attempted that, as Methods to see if a polynomial is irreducible. Although I wonder how much this can be treated as a duplicate since specific polynomial might require different method to be solved and it might not be apparent to which method does the duplicate link refer to...
– Sil
yesterday
add a comment |Â
4 Answers
4
active
oldest
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up vote
4
down vote
accepted
$$p(x)q(x)=(x^2+ax+b)(x^3+cx^2+dx+e)=x^5+2x+1$$
Make the coefficients of same powers on both sides equal.
You will get a system to be studied.
$$c+a=0, d+ac +b=0, e+ad+bc=0,ae+bd=2,be=1$$
We have $$be=1$$ which gives us two possibilities.
$$ b=1,e=1$$ or $$b=-1, e=-1$$
In both cases you will come up with no integer solutions.
answered yesterday
Mohammad Riazi-Kermani
26.9k41849
add a comment |Â
up vote
5
down vote
Looking at questions like these modulo a prime is not guaranteed to work but it is definitely something that should be in your toolbox.
- The theoretical justification comes from a corollary to Gauss' lemma
and states that a monic polynomial $f(x)inBbbZ[x]$ is irreducible
in $BbbQ[x]$ if and only if it is irreducible in $BbbZ[x]$. - Should it happen that $f(x)=g(x)h(x)$ non-trivially in $BbbZ[x]$, then both $g$ and $h$ can be assumed to be monic. Reducing that equation modulo a prime $p$ tells that $overlinef(x)=overlineg(x)overlineh(x)$ in $BbbZ_p[x]$. As $g$ and $h$ were monic, we have $deg g=deg overlineg$ and $deg h=deg overlineh$.
- Checking for irreducibility in $BbbZ_p[x]$ is often easier because, at least in principle, we can try all the alternatives as the field of coefficients is finite. Should it happen that $overlinef$ is irreducible then we can immediately conclude that $f$ is also irreducible.
- Factoring in $BbbZ_p[x]$ is also easier (for small $p$ it is feasible to build a list of low degree irreducible polynomials and use those much the same way you use a list of small primes to factor smallish integers). Even if $overlinef$ is not irreducible we may still get information about the degrees of $g$ and $h$. Behold!
Let's look at your example polynomial $f(x)=x^5+2x+1$ modulo primes $p=2$ and $p=3$. Either choice of $p$ leads to a proof of irreducibility.
- When $p=2$ we get $overlinef=x^5+1=(x+1)(x^4+x^3+x^2+x+1)$. Here it is not immediately obvious that the degree four polynomial is irreducible, but
it does follow from the facts that A) it has no zeros modulo two (only two choices to test!), B) $x^2+x+1$ is the only irreducible quadratic in $BbbZ_2[x]$, and e.g. long division shows that the quartic is not divisible by that either. How does this help us? What we have at this point is that if $f$ is not irreducible it must be a product of a linear factor and a quartic factor. But the rational root test tells you that $f(x)$ does not have a linear factor. We are done - $f$ must be irreducible! Here the key was that a modulo $2$ consideration showed that $f$ cannot have a quadratic factor! - As Lulu pointed out, $f(x)$ is irreducible modulo $p=3$. It is, of course,
easy to see that none of $f(0),f(1),f(2)$ are congruent to $0$ modulo $3$, so $f(x)$ has no linear factors. To exclude the possibility of a quadratic factor takes a bit more work here than it did with $p=2$. Anyway, with a bit of work we can show that $p_1(x)=x^2+1$, $p_2(x)=(x-1)^2+1$ and $p_3(x)=(x+1)^2+1$ are all the irreducible monic quadratics in $BbbZ_3[x]$.
It takes a bit of work to show that none of them are factors of $f(x)$. We can do that in one swoop if we use the fact that
$$(x-1)(x+1)p_1(x)p_2(x)p_3(x)=x^8-1$$
(this comes from a general related result, ask if you want to know more), and then compute with Euclid's algorithm that $gcd(x^8-1,f(x))=1$ in $BbbZ_3[x]$. Anyway, we then know that $f$ is irreducible modulo three, and we are done.
For comparison, the methods in other answers are all very fine. They work well here as well as in many other cases. They work in many cases where modular techniques may fail. But, modular tricks are good to know. For example, instead of your $f(x)$ consider the polynomial
$$
f(x)=x^5+2x+7^n,
$$
where $n$ is either a parameter or some ridiculously large natural number. AFAICT all the other answers fail or lead to uncomfortably many cases. But,
the above proof using reduction modulo $3$ goes through verbatim. The modulo $2$ argument survives if you can do the part with the rational root test and show that $f(pm 7^ell)neq0$ for all $ell, 0leellle n$. One term will usually dominate the others.
answered yesterday
Jyrki Lahtonen
104k12160355
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up vote
4
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The $f(4)=1033$ is a prime, and the polynomial is irreducible by Cohn's irreducibility criterion.
If we invert the polynomial to $g(x)=x^5f(1/x)=x^5+2x^4+1$, we can also use Perron's criterion (can be found in Prasolov's book Polynomials, Theorem 2.2.5):
(Perron's criterion, non-sharp version) Let $f(x)=x^n+a_1x^n-1+dots+a_n$ be a polynomial with integer coefficients such that $a_nneq 0$. If $|a_1| geq 1+|a_2|+dots+|a_n|$ and $f(pm 1)neq 0$, then $f$ is irreducible.
Since $2geq 1+1$, and $f(1)=4neq 0,f(-1)=2neq 0$ and so the polynomial is irreducible.
answered yesterday
Sil
5,06821242
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up vote
2
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A Complex-Analytic Proof
We shall first verify that exactly one root $zetainmathbbC$ of $f(x):=x^5+2x+1$ satisfies $|zeta|<1$. Let $epsilon$ be an arbitrary real number such that $0<epsilon<frac14$. Consider the open ball $B_1-epsilon(0)subseteq mathbbC$ centered at $0$ with radius $1-epsilon$. We see that, for a complex number $z$ on the boundary $partial B_1-epsilon(0)$ of $B_1-epsilon(0)$,
$$|2z|=2(1-epsilon)text and |z^5+1|leq |z|^5+1=(1-epsilon)^5+1,.$$
Observe that
$$beginalign(1-epsilon)^5&=1-5epsilon+epsilon^2(10-10epsilon+5epsilon^2-epsilon^3)
\
&< 1-5epsilon+epsilon^2left(10-0+frac54^2-0right)
\
&<1-5epsilon+11epsilon^2,,endalign$$
as $0<epsilon<frac14$.
That is, $3-11epsilon>0$, and so
$$(1-epsilon)^5<1-5epsilon+11epsilon^2=(1-2epsilon)-epsilon(3-11epsilon)<1-2epsilon,.$$
Consequently,
$$|2z|>left|z^5+1right|$$
for $zinpartial B_1-epsilon(0)$. By Rouché's Theorem, the number of roots of $$f(z)=z^5+2z+1=(2z)+left(z^5+1right)$$ in $B_1-epsilon(0)$ is the same as the number of roots of $2z$ in $B_1-epsilon(0)$, which is $1$. Hence, $f(z)$ has exactly one root $z=zeta$ inside $bigcuplimits_epsiloninleft(0,frac14right),B_1-epsilon(0)=B_1(0)$.
It is also easy to see that $f(x)$ has no root of modulus $1$. This is because $f(r)=0$ with $|r|=1$ implies
$$1=|r|^5=big|r^5big|=|-2r-1|geq 2|-r|-|-1|=2|r|-1=2cdot 1-1=1,,$$
whence we have an equality. By the equality condition of the Triangle Inequality, we must have $r=-1$, but $f(-1)=-2neq 0$. Ergo, the roots of $f(z)$ other than $z=zeta$ have moduli larger than $1$.
Finally, if $f(x)=p(x),q(x)$ for some nonconstant $p(x),q(x)inmathbbZ[x]$, then we can assume without loss of generality that $p(zeta)=0$. Ergo, all the roots of $q(x)$ must have moduli greater than $1$. That is, the constant term of $q(x)$ must be an integer $c$ with $|c|>1$. However, $c$ must divide the constant term of $f(x)$, which is $1$. This is absurd. Therefore, $f(x)$ is an irreducible polynomial over $mathbbZ$ (whence also over $mathbbQ$).
answered yesterday
Batominovski
22.3k22676
@Btominovski Sir. Thank you for your answer. But i didn't learn these concepts, so couldn't understand.
– Identicon
yesterday
1
It is a very useful technique, though. There are IMO problems about integral polynomials that end up using complex numbers. Sometimes, you have a large integral polynomial, and it is easier to learn about the complex roots of this polynomial.
– Batominovski
yesterday
I will learn them soon sir.
– Identicon
yesterday
Sir, Can you send some links to learn about these concepts from basics . Please.
– Identicon
yesterday
1
This is one good pdf file that you can take a look at but it doesn't contain the foundations: yufeizhao.com/olympiad/intpoly.pdf. If you want a solid introduction to complex analysis, you are going to really have to learn it. Search "introduction to complex analysis" or something. There are probably tons of online lecture notes you can read.
– Batominovski
yesterday
 |Â
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
$$p(x)q(x)=(x^2+ax+b)(x^3+cx^2+dx+e)=x^5+2x+1$$
Make the coefficients of same powers on both sides equal.
You will get a system to be studied.
$$c+a=0, d+ac +b=0, e+ad+bc=0,ae+bd=2,be=1$$
We have $$be=1$$ which gives us two possibilities.
$$ b=1,e=1$$ or $$b=-1, e=-1$$
In both cases you will come up with no integer solutions.
answered yesterday
Mohammad Riazi-Kermani
26.9k41849
add a comment |Â
up vote
4
down vote
accepted
$$p(x)q(x)=(x^2+ax+b)(x^3+cx^2+dx+e)=x^5+2x+1$$
Make the coefficients of same powers on both sides equal.
You will get a system to be studied.
$$c+a=0, d+ac +b=0, e+ad+bc=0,ae+bd=2,be=1$$
We have $$be=1$$ which gives us two possibilities.
$$ b=1,e=1$$ or $$b=-1, e=-1$$
In both cases you will come up with no integer solutions.
answered yesterday
Mohammad Riazi-Kermani
26.9k41849
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
$$p(x)q(x)=(x^2+ax+b)(x^3+cx^2+dx+e)=x^5+2x+1$$
Make the coefficients of same powers on both sides equal.
You will get a system to be studied.
$$c+a=0, d+ac +b=0, e+ad+bc=0,ae+bd=2,be=1$$
We have $$be=1$$ which gives us two possibilities.
$$ b=1,e=1$$ or $$b=-1, e=-1$$
In both cases you will come up with no integer solutions.
answered yesterday
Mohammad Riazi-Kermani
26.9k41849
$$p(x)q(x)=(x^2+ax+b)(x^3+cx^2+dx+e)=x^5+2x+1$$
Make the coefficients of same powers on both sides equal.
You will get a system to be studied.
$$c+a=0, d+ac +b=0, e+ad+bc=0,ae+bd=2,be=1$$
We have $$be=1$$ which gives us two possibilities.
$$ b=1,e=1$$ or $$b=-1, e=-1$$
In both cases you will come up with no integer solutions.
answered yesterday
Mohammad Riazi-Kermani
26.9k41849
answered yesterday
Mohammad Riazi-Kermani
26.9k41849
answered yesterday
Mohammad Riazi-Kermani
26.9k41849
answered yesterday
Mohammad Riazi-Kermani
26.9k41849
26.9k41849
add a comment |Â
add a comment |Â
up vote
5
down vote
Looking at questions like these modulo a prime is not guaranteed to work but it is definitely something that should be in your toolbox.
- The theoretical justification comes from a corollary to Gauss' lemma
and states that a monic polynomial $f(x)inBbbZ[x]$ is irreducible
in $BbbQ[x]$ if and only if it is irreducible in $BbbZ[x]$. - Should it happen that $f(x)=g(x)h(x)$ non-trivially in $BbbZ[x]$, then both $g$ and $h$ can be assumed to be monic. Reducing that equation modulo a prime $p$ tells that $overlinef(x)=overlineg(x)overlineh(x)$ in $BbbZ_p[x]$. As $g$ and $h$ were monic, we have $deg g=deg overlineg$ and $deg h=deg overlineh$.
- Checking for irreducibility in $BbbZ_p[x]$ is often easier because, at least in principle, we can try all the alternatives as the field of coefficients is finite. Should it happen that $overlinef$ is irreducible then we can immediately conclude that $f$ is also irreducible.
- Factoring in $BbbZ_p[x]$ is also easier (for small $p$ it is feasible to build a list of low degree irreducible polynomials and use those much the same way you use a list of small primes to factor smallish integers). Even if $overlinef$ is not irreducible we may still get information about the degrees of $g$ and $h$. Behold!
Let's look at your example polynomial $f(x)=x^5+2x+1$ modulo primes $p=2$ and $p=3$. Either choice of $p$ leads to a proof of irreducibility.
- When $p=2$ we get $overlinef=x^5+1=(x+1)(x^4+x^3+x^2+x+1)$. Here it is not immediately obvious that the degree four polynomial is irreducible, but
it does follow from the facts that A) it has no zeros modulo two (only two choices to test!), B) $x^2+x+1$ is the only irreducible quadratic in $BbbZ_2[x]$, and e.g. long division shows that the quartic is not divisible by that either. How does this help us? What we have at this point is that if $f$ is not irreducible it must be a product of a linear factor and a quartic factor. But the rational root test tells you that $f(x)$ does not have a linear factor. We are done - $f$ must be irreducible! Here the key was that a modulo $2$ consideration showed that $f$ cannot have a quadratic factor! - As Lulu pointed out, $f(x)$ is irreducible modulo $p=3$. It is, of course,
easy to see that none of $f(0),f(1),f(2)$ are congruent to $0$ modulo $3$, so $f(x)$ has no linear factors. To exclude the possibility of a quadratic factor takes a bit more work here than it did with $p=2$. Anyway, with a bit of work we can show that $p_1(x)=x^2+1$, $p_2(x)=(x-1)^2+1$ and $p_3(x)=(x+1)^2+1$ are all the irreducible monic quadratics in $BbbZ_3[x]$.
It takes a bit of work to show that none of them are factors of $f(x)$. We can do that in one swoop if we use the fact that
$$(x-1)(x+1)p_1(x)p_2(x)p_3(x)=x^8-1$$
(this comes from a general related result, ask if you want to know more), and then compute with Euclid's algorithm that $gcd(x^8-1,f(x))=1$ in $BbbZ_3[x]$. Anyway, we then know that $f$ is irreducible modulo three, and we are done.
For comparison, the methods in other answers are all very fine. They work well here as well as in many other cases. They work in many cases where modular techniques may fail. But, modular tricks are good to know. For example, instead of your $f(x)$ consider the polynomial
$$
f(x)=x^5+2x+7^n,
$$
where $n$ is either a parameter or some ridiculously large natural number. AFAICT all the other answers fail or lead to uncomfortably many cases. But,
the above proof using reduction modulo $3$ goes through verbatim. The modulo $2$ argument survives if you can do the part with the rational root test and show that $f(pm 7^ell)neq0$ for all $ell, 0leellle n$. One term will usually dominate the others.
answered yesterday
Jyrki Lahtonen
104k12160355
add a comment |Â
up vote
5
down vote
Looking at questions like these modulo a prime is not guaranteed to work but it is definitely something that should be in your toolbox.
- The theoretical justification comes from a corollary to Gauss' lemma
and states that a monic polynomial $f(x)inBbbZ[x]$ is irreducible
in $BbbQ[x]$ if and only if it is irreducible in $BbbZ[x]$. - Should it happen that $f(x)=g(x)h(x)$ non-trivially in $BbbZ[x]$, then both $g$ and $h$ can be assumed to be monic. Reducing that equation modulo a prime $p$ tells that $overlinef(x)=overlineg(x)overlineh(x)$ in $BbbZ_p[x]$. As $g$ and $h$ were monic, we have $deg g=deg overlineg$ and $deg h=deg overlineh$.
- Checking for irreducibility in $BbbZ_p[x]$ is often easier because, at least in principle, we can try all the alternatives as the field of coefficients is finite. Should it happen that $overlinef$ is irreducible then we can immediately conclude that $f$ is also irreducible.
- Factoring in $BbbZ_p[x]$ is also easier (for small $p$ it is feasible to build a list of low degree irreducible polynomials and use those much the same way you use a list of small primes to factor smallish integers). Even if $overlinef$ is not irreducible we may still get information about the degrees of $g$ and $h$. Behold!
Let's look at your example polynomial $f(x)=x^5+2x+1$ modulo primes $p=2$ and $p=3$. Either choice of $p$ leads to a proof of irreducibility.
- When $p=2$ we get $overlinef=x^5+1=(x+1)(x^4+x^3+x^2+x+1)$. Here it is not immediately obvious that the degree four polynomial is irreducible, but
it does follow from the facts that A) it has no zeros modulo two (only two choices to test!), B) $x^2+x+1$ is the only irreducible quadratic in $BbbZ_2[x]$, and e.g. long division shows that the quartic is not divisible by that either. How does this help us? What we have at this point is that if $f$ is not irreducible it must be a product of a linear factor and a quartic factor. But the rational root test tells you that $f(x)$ does not have a linear factor. We are done - $f$ must be irreducible! Here the key was that a modulo $2$ consideration showed that $f$ cannot have a quadratic factor! - As Lulu pointed out, $f(x)$ is irreducible modulo $p=3$. It is, of course,
easy to see that none of $f(0),f(1),f(2)$ are congruent to $0$ modulo $3$, so $f(x)$ has no linear factors. To exclude the possibility of a quadratic factor takes a bit more work here than it did with $p=2$. Anyway, with a bit of work we can show that $p_1(x)=x^2+1$, $p_2(x)=(x-1)^2+1$ and $p_3(x)=(x+1)^2+1$ are all the irreducible monic quadratics in $BbbZ_3[x]$.
It takes a bit of work to show that none of them are factors of $f(x)$. We can do that in one swoop if we use the fact that
$$(x-1)(x+1)p_1(x)p_2(x)p_3(x)=x^8-1$$
(this comes from a general related result, ask if you want to know more), and then compute with Euclid's algorithm that $gcd(x^8-1,f(x))=1$ in $BbbZ_3[x]$. Anyway, we then know that $f$ is irreducible modulo three, and we are done.
For comparison, the methods in other answers are all very fine. They work well here as well as in many other cases. They work in many cases where modular techniques may fail. But, modular tricks are good to know. For example, instead of your $f(x)$ consider the polynomial
$$
f(x)=x^5+2x+7^n,
$$
where $n$ is either a parameter or some ridiculously large natural number. AFAICT all the other answers fail or lead to uncomfortably many cases. But,
the above proof using reduction modulo $3$ goes through verbatim. The modulo $2$ argument survives if you can do the part with the rational root test and show that $f(pm 7^ell)neq0$ for all $ell, 0leellle n$. One term will usually dominate the others.
answered yesterday
Jyrki Lahtonen
104k12160355
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Looking at questions like these modulo a prime is not guaranteed to work but it is definitely something that should be in your toolbox.
- The theoretical justification comes from a corollary to Gauss' lemma
and states that a monic polynomial $f(x)inBbbZ[x]$ is irreducible
in $BbbQ[x]$ if and only if it is irreducible in $BbbZ[x]$. - Should it happen that $f(x)=g(x)h(x)$ non-trivially in $BbbZ[x]$, then both $g$ and $h$ can be assumed to be monic. Reducing that equation modulo a prime $p$ tells that $overlinef(x)=overlineg(x)overlineh(x)$ in $BbbZ_p[x]$. As $g$ and $h$ were monic, we have $deg g=deg overlineg$ and $deg h=deg overlineh$.
- Checking for irreducibility in $BbbZ_p[x]$ is often easier because, at least in principle, we can try all the alternatives as the field of coefficients is finite. Should it happen that $overlinef$ is irreducible then we can immediately conclude that $f$ is also irreducible.
- Factoring in $BbbZ_p[x]$ is also easier (for small $p$ it is feasible to build a list of low degree irreducible polynomials and use those much the same way you use a list of small primes to factor smallish integers). Even if $overlinef$ is not irreducible we may still get information about the degrees of $g$ and $h$. Behold!
Let's look at your example polynomial $f(x)=x^5+2x+1$ modulo primes $p=2$ and $p=3$. Either choice of $p$ leads to a proof of irreducibility.
- When $p=2$ we get $overlinef=x^5+1=(x+1)(x^4+x^3+x^2+x+1)$. Here it is not immediately obvious that the degree four polynomial is irreducible, but
it does follow from the facts that A) it has no zeros modulo two (only two choices to test!), B) $x^2+x+1$ is the only irreducible quadratic in $BbbZ_2[x]$, and e.g. long division shows that the quartic is not divisible by that either. How does this help us? What we have at this point is that if $f$ is not irreducible it must be a product of a linear factor and a quartic factor. But the rational root test tells you that $f(x)$ does not have a linear factor. We are done - $f$ must be irreducible! Here the key was that a modulo $2$ consideration showed that $f$ cannot have a quadratic factor! - As Lulu pointed out, $f(x)$ is irreducible modulo $p=3$. It is, of course,
easy to see that none of $f(0),f(1),f(2)$ are congruent to $0$ modulo $3$, so $f(x)$ has no linear factors. To exclude the possibility of a quadratic factor takes a bit more work here than it did with $p=2$. Anyway, with a bit of work we can show that $p_1(x)=x^2+1$, $p_2(x)=(x-1)^2+1$ and $p_3(x)=(x+1)^2+1$ are all the irreducible monic quadratics in $BbbZ_3[x]$.
It takes a bit of work to show that none of them are factors of $f(x)$. We can do that in one swoop if we use the fact that
$$(x-1)(x+1)p_1(x)p_2(x)p_3(x)=x^8-1$$
(this comes from a general related result, ask if you want to know more), and then compute with Euclid's algorithm that $gcd(x^8-1,f(x))=1$ in $BbbZ_3[x]$. Anyway, we then know that $f$ is irreducible modulo three, and we are done.
For comparison, the methods in other answers are all very fine. They work well here as well as in many other cases. They work in many cases where modular techniques may fail. But, modular tricks are good to know. For example, instead of your $f(x)$ consider the polynomial
$$
f(x)=x^5+2x+7^n,
$$
where $n$ is either a parameter or some ridiculously large natural number. AFAICT all the other answers fail or lead to uncomfortably many cases. But,
the above proof using reduction modulo $3$ goes through verbatim. The modulo $2$ argument survives if you can do the part with the rational root test and show that $f(pm 7^ell)neq0$ for all $ell, 0leellle n$. One term will usually dominate the others.
answered yesterday
Jyrki Lahtonen
104k12160355
Looking at questions like these modulo a prime is not guaranteed to work but it is definitely something that should be in your toolbox.
- The theoretical justification comes from a corollary to Gauss' lemma
and states that a monic polynomial $f(x)inBbbZ[x]$ is irreducible
in $BbbQ[x]$ if and only if it is irreducible in $BbbZ[x]$. - Should it happen that $f(x)=g(x)h(x)$ non-trivially in $BbbZ[x]$, then both $g$ and $h$ can be assumed to be monic. Reducing that equation modulo a prime $p$ tells that $overlinef(x)=overlineg(x)overlineh(x)$ in $BbbZ_p[x]$. As $g$ and $h$ were monic, we have $deg g=deg overlineg$ and $deg h=deg overlineh$.
- Checking for irreducibility in $BbbZ_p[x]$ is often easier because, at least in principle, we can try all the alternatives as the field of coefficients is finite. Should it happen that $overlinef$ is irreducible then we can immediately conclude that $f$ is also irreducible.
- Factoring in $BbbZ_p[x]$ is also easier (for small $p$ it is feasible to build a list of low degree irreducible polynomials and use those much the same way you use a list of small primes to factor smallish integers). Even if $overlinef$ is not irreducible we may still get information about the degrees of $g$ and $h$. Behold!
Let's look at your example polynomial $f(x)=x^5+2x+1$ modulo primes $p=2$ and $p=3$. Either choice of $p$ leads to a proof of irreducibility.
- When $p=2$ we get $overlinef=x^5+1=(x+1)(x^4+x^3+x^2+x+1)$. Here it is not immediately obvious that the degree four polynomial is irreducible, but
it does follow from the facts that A) it has no zeros modulo two (only two choices to test!), B) $x^2+x+1$ is the only irreducible quadratic in $BbbZ_2[x]$, and e.g. long division shows that the quartic is not divisible by that either. How does this help us? What we have at this point is that if $f$ is not irreducible it must be a product of a linear factor and a quartic factor. But the rational root test tells you that $f(x)$ does not have a linear factor. We are done - $f$ must be irreducible! Here the key was that a modulo $2$ consideration showed that $f$ cannot have a quadratic factor! - As Lulu pointed out, $f(x)$ is irreducible modulo $p=3$. It is, of course,
easy to see that none of $f(0),f(1),f(2)$ are congruent to $0$ modulo $3$, so $f(x)$ has no linear factors. To exclude the possibility of a quadratic factor takes a bit more work here than it did with $p=2$. Anyway, with a bit of work we can show that $p_1(x)=x^2+1$, $p_2(x)=(x-1)^2+1$ and $p_3(x)=(x+1)^2+1$ are all the irreducible monic quadratics in $BbbZ_3[x]$.
It takes a bit of work to show that none of them are factors of $f(x)$. We can do that in one swoop if we use the fact that
$$(x-1)(x+1)p_1(x)p_2(x)p_3(x)=x^8-1$$
(this comes from a general related result, ask if you want to know more), and then compute with Euclid's algorithm that $gcd(x^8-1,f(x))=1$ in $BbbZ_3[x]$. Anyway, we then know that $f$ is irreducible modulo three, and we are done.
For comparison, the methods in other answers are all very fine. They work well here as well as in many other cases. They work in many cases where modular techniques may fail. But, modular tricks are good to know. For example, instead of your $f(x)$ consider the polynomial
$$
f(x)=x^5+2x+7^n,
$$
where $n$ is either a parameter or some ridiculously large natural number. AFAICT all the other answers fail or lead to uncomfortably many cases. But,
the above proof using reduction modulo $3$ goes through verbatim. The modulo $2$ argument survives if you can do the part with the rational root test and show that $f(pm 7^ell)neq0$ for all $ell, 0leellle n$. One term will usually dominate the others.
answered yesterday
Jyrki Lahtonen
104k12160355
edited yesterday
answered yesterday
Jyrki Lahtonen
104k12160355
answered yesterday
Jyrki Lahtonen
104k12160355
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104k12160355
104k12160355
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The $f(4)=1033$ is a prime, and the polynomial is irreducible by Cohn's irreducibility criterion.
If we invert the polynomial to $g(x)=x^5f(1/x)=x^5+2x^4+1$, we can also use Perron's criterion (can be found in Prasolov's book Polynomials, Theorem 2.2.5):
(Perron's criterion, non-sharp version) Let $f(x)=x^n+a_1x^n-1+dots+a_n$ be a polynomial with integer coefficients such that $a_nneq 0$. If $|a_1| geq 1+|a_2|+dots+|a_n|$ and $f(pm 1)neq 0$, then $f$ is irreducible.
Since $2geq 1+1$, and $f(1)=4neq 0,f(-1)=2neq 0$ and so the polynomial is irreducible.
answered yesterday
Sil
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The $f(4)=1033$ is a prime, and the polynomial is irreducible by Cohn's irreducibility criterion.
If we invert the polynomial to $g(x)=x^5f(1/x)=x^5+2x^4+1$, we can also use Perron's criterion (can be found in Prasolov's book Polynomials, Theorem 2.2.5):
(Perron's criterion, non-sharp version) Let $f(x)=x^n+a_1x^n-1+dots+a_n$ be a polynomial with integer coefficients such that $a_nneq 0$. If $|a_1| geq 1+|a_2|+dots+|a_n|$ and $f(pm 1)neq 0$, then $f$ is irreducible.
Since $2geq 1+1$, and $f(1)=4neq 0,f(-1)=2neq 0$ and so the polynomial is irreducible.
answered yesterday
Sil
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up vote
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The $f(4)=1033$ is a prime, and the polynomial is irreducible by Cohn's irreducibility criterion.
If we invert the polynomial to $g(x)=x^5f(1/x)=x^5+2x^4+1$, we can also use Perron's criterion (can be found in Prasolov's book Polynomials, Theorem 2.2.5):
(Perron's criterion, non-sharp version) Let $f(x)=x^n+a_1x^n-1+dots+a_n$ be a polynomial with integer coefficients such that $a_nneq 0$. If $|a_1| geq 1+|a_2|+dots+|a_n|$ and $f(pm 1)neq 0$, then $f$ is irreducible.
Since $2geq 1+1$, and $f(1)=4neq 0,f(-1)=2neq 0$ and so the polynomial is irreducible.
answered yesterday
Sil
5,06821242
The $f(4)=1033$ is a prime, and the polynomial is irreducible by Cohn's irreducibility criterion.
If we invert the polynomial to $g(x)=x^5f(1/x)=x^5+2x^4+1$, we can also use Perron's criterion (can be found in Prasolov's book Polynomials, Theorem 2.2.5):
(Perron's criterion, non-sharp version) Let $f(x)=x^n+a_1x^n-1+dots+a_n$ be a polynomial with integer coefficients such that $a_nneq 0$. If $|a_1| geq 1+|a_2|+dots+|a_n|$ and $f(pm 1)neq 0$, then $f$ is irreducible.
Since $2geq 1+1$, and $f(1)=4neq 0,f(-1)=2neq 0$ and so the polynomial is irreducible.
answered yesterday
Sil
5,06821242
answered yesterday
Sil
5,06821242
answered yesterday
Sil
5,06821242
answered yesterday
Sil
5,06821242
5,06821242
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A Complex-Analytic Proof
We shall first verify that exactly one root $zetainmathbbC$ of $f(x):=x^5+2x+1$ satisfies $|zeta|<1$. Let $epsilon$ be an arbitrary real number such that $0<epsilon<frac14$. Consider the open ball $B_1-epsilon(0)subseteq mathbbC$ centered at $0$ with radius $1-epsilon$. We see that, for a complex number $z$ on the boundary $partial B_1-epsilon(0)$ of $B_1-epsilon(0)$,
$$|2z|=2(1-epsilon)text and |z^5+1|leq |z|^5+1=(1-epsilon)^5+1,.$$
Observe that
$$beginalign(1-epsilon)^5&=1-5epsilon+epsilon^2(10-10epsilon+5epsilon^2-epsilon^3)
\
&< 1-5epsilon+epsilon^2left(10-0+frac54^2-0right)
\
&<1-5epsilon+11epsilon^2,,endalign$$
as $0<epsilon<frac14$.
That is, $3-11epsilon>0$, and so
$$(1-epsilon)^5<1-5epsilon+11epsilon^2=(1-2epsilon)-epsilon(3-11epsilon)<1-2epsilon,.$$
Consequently,
$$|2z|>left|z^5+1right|$$
for $zinpartial B_1-epsilon(0)$. By Rouché's Theorem, the number of roots of $$f(z)=z^5+2z+1=(2z)+left(z^5+1right)$$ in $B_1-epsilon(0)$ is the same as the number of roots of $2z$ in $B_1-epsilon(0)$, which is $1$. Hence, $f(z)$ has exactly one root $z=zeta$ inside $bigcuplimits_epsiloninleft(0,frac14right),B_1-epsilon(0)=B_1(0)$.
It is also easy to see that $f(x)$ has no root of modulus $1$. This is because $f(r)=0$ with $|r|=1$ implies
$$1=|r|^5=big|r^5big|=|-2r-1|geq 2|-r|-|-1|=2|r|-1=2cdot 1-1=1,,$$
whence we have an equality. By the equality condition of the Triangle Inequality, we must have $r=-1$, but $f(-1)=-2neq 0$. Ergo, the roots of $f(z)$ other than $z=zeta$ have moduli larger than $1$.
Finally, if $f(x)=p(x),q(x)$ for some nonconstant $p(x),q(x)inmathbbZ[x]$, then we can assume without loss of generality that $p(zeta)=0$. Ergo, all the roots of $q(x)$ must have moduli greater than $1$. That is, the constant term of $q(x)$ must be an integer $c$ with $|c|>1$. However, $c$ must divide the constant term of $f(x)$, which is $1$. This is absurd. Therefore, $f(x)$ is an irreducible polynomial over $mathbbZ$ (whence also over $mathbbQ$).
answered yesterday
Batominovski
22.3k22676
@Btominovski Sir. Thank you for your answer. But i didn't learn these concepts, so couldn't understand.
– Identicon
yesterday
1
It is a very useful technique, though. There are IMO problems about integral polynomials that end up using complex numbers. Sometimes, you have a large integral polynomial, and it is easier to learn about the complex roots of this polynomial.
– Batominovski
yesterday
I will learn them soon sir.
– Identicon
yesterday
Sir, Can you send some links to learn about these concepts from basics . Please.
– Identicon
yesterday
1
This is one good pdf file that you can take a look at but it doesn't contain the foundations: yufeizhao.com/olympiad/intpoly.pdf. If you want a solid introduction to complex analysis, you are going to really have to learn it. Search "introduction to complex analysis" or something. There are probably tons of online lecture notes you can read.
– Batominovski
yesterday
 |Â
show 1 more comment
up vote
2
down vote
A Complex-Analytic Proof
We shall first verify that exactly one root $zetainmathbbC$ of $f(x):=x^5+2x+1$ satisfies $|zeta|<1$. Let $epsilon$ be an arbitrary real number such that $0<epsilon<frac14$. Consider the open ball $B_1-epsilon(0)subseteq mathbbC$ centered at $0$ with radius $1-epsilon$. We see that, for a complex number $z$ on the boundary $partial B_1-epsilon(0)$ of $B_1-epsilon(0)$,
$$|2z|=2(1-epsilon)text and |z^5+1|leq |z|^5+1=(1-epsilon)^5+1,.$$
Observe that
$$beginalign(1-epsilon)^5&=1-5epsilon+epsilon^2(10-10epsilon+5epsilon^2-epsilon^3)
\
&< 1-5epsilon+epsilon^2left(10-0+frac54^2-0right)
\
&<1-5epsilon+11epsilon^2,,endalign$$
as $0<epsilon<frac14$.
That is, $3-11epsilon>0$, and so
$$(1-epsilon)^5<1-5epsilon+11epsilon^2=(1-2epsilon)-epsilon(3-11epsilon)<1-2epsilon,.$$
Consequently,
$$|2z|>left|z^5+1right|$$
for $zinpartial B_1-epsilon(0)$. By Rouché's Theorem, the number of roots of $$f(z)=z^5+2z+1=(2z)+left(z^5+1right)$$ in $B_1-epsilon(0)$ is the same as the number of roots of $2z$ in $B_1-epsilon(0)$, which is $1$. Hence, $f(z)$ has exactly one root $z=zeta$ inside $bigcuplimits_epsiloninleft(0,frac14right),B_1-epsilon(0)=B_1(0)$.
It is also easy to see that $f(x)$ has no root of modulus $1$. This is because $f(r)=0$ with $|r|=1$ implies
$$1=|r|^5=big|r^5big|=|-2r-1|geq 2|-r|-|-1|=2|r|-1=2cdot 1-1=1,,$$
whence we have an equality. By the equality condition of the Triangle Inequality, we must have $r=-1$, but $f(-1)=-2neq 0$. Ergo, the roots of $f(z)$ other than $z=zeta$ have moduli larger than $1$.
Finally, if $f(x)=p(x),q(x)$ for some nonconstant $p(x),q(x)inmathbbZ[x]$, then we can assume without loss of generality that $p(zeta)=0$. Ergo, all the roots of $q(x)$ must have moduli greater than $1$. That is, the constant term of $q(x)$ must be an integer $c$ with $|c|>1$. However, $c$ must divide the constant term of $f(x)$, which is $1$. This is absurd. Therefore, $f(x)$ is an irreducible polynomial over $mathbbZ$ (whence also over $mathbbQ$).
answered yesterday
Batominovski
22.3k22676
@Btominovski Sir. Thank you for your answer. But i didn't learn these concepts, so couldn't understand.
– Identicon
yesterday
1
It is a very useful technique, though. There are IMO problems about integral polynomials that end up using complex numbers. Sometimes, you have a large integral polynomial, and it is easier to learn about the complex roots of this polynomial.
– Batominovski
yesterday
I will learn them soon sir.
– Identicon
yesterday
Sir, Can you send some links to learn about these concepts from basics . Please.
– Identicon
yesterday
1
This is one good pdf file that you can take a look at but it doesn't contain the foundations: yufeizhao.com/olympiad/intpoly.pdf. If you want a solid introduction to complex analysis, you are going to really have to learn it. Search "introduction to complex analysis" or something. There are probably tons of online lecture notes you can read.
– Batominovski
yesterday
 |Â
show 1 more comment
up vote
2
down vote
up vote
2
down vote
A Complex-Analytic Proof
We shall first verify that exactly one root $zetainmathbbC$ of $f(x):=x^5+2x+1$ satisfies $|zeta|<1$. Let $epsilon$ be an arbitrary real number such that $0<epsilon<frac14$. Consider the open ball $B_1-epsilon(0)subseteq mathbbC$ centered at $0$ with radius $1-epsilon$. We see that, for a complex number $z$ on the boundary $partial B_1-epsilon(0)$ of $B_1-epsilon(0)$,
$$|2z|=2(1-epsilon)text and |z^5+1|leq |z|^5+1=(1-epsilon)^5+1,.$$
Observe that
$$beginalign(1-epsilon)^5&=1-5epsilon+epsilon^2(10-10epsilon+5epsilon^2-epsilon^3)
\
&< 1-5epsilon+epsilon^2left(10-0+frac54^2-0right)
\
&<1-5epsilon+11epsilon^2,,endalign$$
as $0<epsilon<frac14$.
That is, $3-11epsilon>0$, and so
$$(1-epsilon)^5<1-5epsilon+11epsilon^2=(1-2epsilon)-epsilon(3-11epsilon)<1-2epsilon,.$$
Consequently,
$$|2z|>left|z^5+1right|$$
for $zinpartial B_1-epsilon(0)$. By Rouché's Theorem, the number of roots of $$f(z)=z^5+2z+1=(2z)+left(z^5+1right)$$ in $B_1-epsilon(0)$ is the same as the number of roots of $2z$ in $B_1-epsilon(0)$, which is $1$. Hence, $f(z)$ has exactly one root $z=zeta$ inside $bigcuplimits_epsiloninleft(0,frac14right),B_1-epsilon(0)=B_1(0)$.
It is also easy to see that $f(x)$ has no root of modulus $1$. This is because $f(r)=0$ with $|r|=1$ implies
$$1=|r|^5=big|r^5big|=|-2r-1|geq 2|-r|-|-1|=2|r|-1=2cdot 1-1=1,,$$
whence we have an equality. By the equality condition of the Triangle Inequality, we must have $r=-1$, but $f(-1)=-2neq 0$. Ergo, the roots of $f(z)$ other than $z=zeta$ have moduli larger than $1$.
Finally, if $f(x)=p(x),q(x)$ for some nonconstant $p(x),q(x)inmathbbZ[x]$, then we can assume without loss of generality that $p(zeta)=0$. Ergo, all the roots of $q(x)$ must have moduli greater than $1$. That is, the constant term of $q(x)$ must be an integer $c$ with $|c|>1$. However, $c$ must divide the constant term of $f(x)$, which is $1$. This is absurd. Therefore, $f(x)$ is an irreducible polynomial over $mathbbZ$ (whence also over $mathbbQ$).
answered yesterday
Batominovski
22.3k22676
A Complex-Analytic Proof
We shall first verify that exactly one root $zetainmathbbC$ of $f(x):=x^5+2x+1$ satisfies $|zeta|<1$. Let $epsilon$ be an arbitrary real number such that $0<epsilon<frac14$. Consider the open ball $B_1-epsilon(0)subseteq mathbbC$ centered at $0$ with radius $1-epsilon$. We see that, for a complex number $z$ on the boundary $partial B_1-epsilon(0)$ of $B_1-epsilon(0)$,
$$|2z|=2(1-epsilon)text and |z^5+1|leq |z|^5+1=(1-epsilon)^5+1,.$$
Observe that
$$beginalign(1-epsilon)^5&=1-5epsilon+epsilon^2(10-10epsilon+5epsilon^2-epsilon^3)
\
&< 1-5epsilon+epsilon^2left(10-0+frac54^2-0right)
\
&<1-5epsilon+11epsilon^2,,endalign$$
as $0<epsilon<frac14$.
That is, $3-11epsilon>0$, and so
$$(1-epsilon)^5<1-5epsilon+11epsilon^2=(1-2epsilon)-epsilon(3-11epsilon)<1-2epsilon,.$$
Consequently,
$$|2z|>left|z^5+1right|$$
for $zinpartial B_1-epsilon(0)$. By Rouché's Theorem, the number of roots of $$f(z)=z^5+2z+1=(2z)+left(z^5+1right)$$ in $B_1-epsilon(0)$ is the same as the number of roots of $2z$ in $B_1-epsilon(0)$, which is $1$. Hence, $f(z)$ has exactly one root $z=zeta$ inside $bigcuplimits_epsiloninleft(0,frac14right),B_1-epsilon(0)=B_1(0)$.
It is also easy to see that $f(x)$ has no root of modulus $1$. This is because $f(r)=0$ with $|r|=1$ implies
$$1=|r|^5=big|r^5big|=|-2r-1|geq 2|-r|-|-1|=2|r|-1=2cdot 1-1=1,,$$
whence we have an equality. By the equality condition of the Triangle Inequality, we must have $r=-1$, but $f(-1)=-2neq 0$. Ergo, the roots of $f(z)$ other than $z=zeta$ have moduli larger than $1$.
Finally, if $f(x)=p(x),q(x)$ for some nonconstant $p(x),q(x)inmathbbZ[x]$, then we can assume without loss of generality that $p(zeta)=0$. Ergo, all the roots of $q(x)$ must have moduli greater than $1$. That is, the constant term of $q(x)$ must be an integer $c$ with $|c|>1$. However, $c$ must divide the constant term of $f(x)$, which is $1$. This is absurd. Therefore, $f(x)$ is an irreducible polynomial over $mathbbZ$ (whence also over $mathbbQ$).
answered yesterday
Batominovski
22.3k22676
edited yesterday
answered yesterday
Batominovski
22.3k22676
answered yesterday
Batominovski
22.3k22676
answered yesterday
Batominovski
22.3k22676
22.3k22676
@Btominovski Sir. Thank you for your answer. But i didn't learn these concepts, so couldn't understand.
– Identicon
yesterday
1
It is a very useful technique, though. There are IMO problems about integral polynomials that end up using complex numbers. Sometimes, you have a large integral polynomial, and it is easier to learn about the complex roots of this polynomial.
– Batominovski
yesterday
I will learn them soon sir.
– Identicon
yesterday
Sir, Can you send some links to learn about these concepts from basics . Please.
– Identicon
yesterday
1
This is one good pdf file that you can take a look at but it doesn't contain the foundations: yufeizhao.com/olympiad/intpoly.pdf. If you want a solid introduction to complex analysis, you are going to really have to learn it. Search "introduction to complex analysis" or something. There are probably tons of online lecture notes you can read.
– Batominovski
yesterday
 |Â
show 1 more comment
@Btominovski Sir. Thank you for your answer. But i didn't learn these concepts, so couldn't understand.
– Identicon
yesterday
1
It is a very useful technique, though. There are IMO problems about integral polynomials that end up using complex numbers. Sometimes, you have a large integral polynomial, and it is easier to learn about the complex roots of this polynomial.
– Batominovski
yesterday
I will learn them soon sir.
– Identicon
yesterday
Sir, Can you send some links to learn about these concepts from basics . Please.
– Identicon
yesterday
1
This is one good pdf file that you can take a look at but it doesn't contain the foundations: yufeizhao.com/olympiad/intpoly.pdf. If you want a solid introduction to complex analysis, you are going to really have to learn it. Search "introduction to complex analysis" or something. There are probably tons of online lecture notes you can read.
– Batominovski
yesterday
@Btominovski Sir. Thank you for your answer. But i didn't learn these concepts, so couldn't understand.
– Identicon
yesterday
@Btominovski Sir. Thank you for your answer. But i didn't learn these concepts, so couldn't understand.
– Identicon
yesterday
1
1
It is a very useful technique, though. There are IMO problems about integral polynomials that end up using complex numbers. Sometimes, you have a large integral polynomial, and it is easier to learn about the complex roots of this polynomial.
– Batominovski
yesterday
It is a very useful technique, though. There are IMO problems about integral polynomials that end up using complex numbers. Sometimes, you have a large integral polynomial, and it is easier to learn about the complex roots of this polynomial.
– Batominovski
yesterday
I will learn them soon sir.
– Identicon
yesterday
I will learn them soon sir.
– Identicon
yesterday
Sir, Can you send some links to learn about these concepts from basics . Please.
– Identicon
yesterday
Sir, Can you send some links to learn about these concepts from basics . Please.
– Identicon
yesterday
1
1
This is one good pdf file that you can take a look at but it doesn't contain the foundations: yufeizhao.com/olympiad/intpoly.pdf. If you want a solid introduction to complex analysis, you are going to really have to learn it. Search "introduction to complex analysis" or something. There are probably tons of online lecture notes you can read.
– Batominovski
yesterday
This is one good pdf file that you can take a look at but it doesn't contain the foundations: yufeizhao.com/olympiad/intpoly.pdf. If you want a solid introduction to complex analysis, you are going to really have to learn it. Search "introduction to complex analysis" or something. There are probably tons of online lecture notes you can read.
– Batominovski
yesterday
 |Â
show 1 more comment
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5
Often it is good to check $pmod p$ for small primes $p$, just because the algebra is so much easier. In this case, the polynomial is irreducible $pmod 3$.
– lulu
yesterday
1
$c+a=0$ because no $x^4.$ Substitute back $c = -a$ and start again. What is the coefficient of $x^3 ; ?$
– Will Jagy
yesterday
1
Two choices, either $b=e=1$ or $b=e=-1.$ There is enough information to finish both cases using $c=-a$ and finding contradictions for each, after finding $d ; ; .$
– Will Jagy
yesterday
1
+1 to all. But I think we should compose a thread covering a compendium of techniques. This types of questions, while not as numerous as those of $a^bmod n$ or integration by partial fraction or..., are still frequent enough that it may serve a point to prepare an abstract duplicate.
– Jyrki Lahtonen
yesterday
@JyrkiLahtonen Like the idea, there are some questions that somehow attempted that, as Methods to see if a polynomial is irreducible. Although I wonder how much this can be treated as a duplicate since specific polynomial might require different method to be solved and it might not be apparent to which method does the duplicate link refer to...
– Sil
yesterday