Irreducibility of a simple polynomial Unicorn Meta Zoo #1: Why another podcast? Announcing the arrival of Valued Associate #679: Cesar ManaraShow $x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $mathbb Q[x]$.Determine whether the polynomial $x^2-12$ in $mathbb Z[x]$ satisfies an Eisenstein criterion for irreducibility over $mathbb Q$Proving Irreducibility of $x^4-16x^3+20x^2+12$ in $mathbb Q[x]$Constructibility of roots of a polynomialEisenstein's criterion for polynomials in Z mod pProving irreducibility; What is this method and what is the logic behind it?Irreducibility of special cyclotomic polynomial.Irreducibility of a Polynomial after a substitutionIrreducibility of Non-monic Quartic Polynomials in Q[x]Irreducible monic polynomial in $mathbbQ[x]$

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Irreducibility of a simple polynomial



Unicorn Meta Zoo #1: Why another podcast?
Announcing the arrival of Valued Associate #679: Cesar ManaraShow $x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $mathbb Q[x]$.Determine whether the polynomial $x^2-12$ in $mathbb Z[x]$ satisfies an Eisenstein criterion for irreducibility over $mathbb Q$Proving Irreducibility of $x^4-16x^3+20x^2+12$ in $mathbb Q[x]$Constructibility of roots of a polynomialEisenstein's criterion for polynomials in Z mod pProving irreducibility; What is this method and what is the logic behind it?Irreducibility of special cyclotomic polynomial.Irreducibility of a Polynomial after a substitutionIrreducibility of Non-monic Quartic Polynomials in Q[x]Irreducible monic polynomial in $mathbbQ[x]$










3












$begingroup$


For an integer $a$, I'm trying to find a criterion to tell me if $x^4+a^2$ is irreducible over $mathbbQ$.



What I've done so far is shown that if $a$ is odd, then $a^2$ is congruent to $1 mod 4$, and so the Eisenstein criterion with $p = 2$ tells me that $(x+1)^4 + a^2$ is irreducible and hence $x^4+a^2$ is irreducible.



For $a$ even I have had no such luck. I know that the polynomial is not irreducible for all such $a$ because when $a = 2$, for example, we have the factorization $x^4+4 = (x^2-2x+2)(x^2+2x+2)$. It's easy to show that this polynomial has no linear factors, but I'm at a loss trying to decide when there are a pair of irreducible quadratic factors.



A thought I had was to try and consider when $mathbbQ(sqrtai)$ is a degree $4$ extension, but this didn't seem to help.










share|cite|improve this question









$endgroup$







  • 2




    $begingroup$
    If $a=2n^2$, then $x^4+a^2=x^4+(2n^2)^2=(x^2-2nx+2n^2)(x^2+2nx+2n^2)$. In other cases it seems to be irreducible.
    $endgroup$
    – Sil
    Mar 26 at 21:08
















3












$begingroup$


For an integer $a$, I'm trying to find a criterion to tell me if $x^4+a^2$ is irreducible over $mathbbQ$.



What I've done so far is shown that if $a$ is odd, then $a^2$ is congruent to $1 mod 4$, and so the Eisenstein criterion with $p = 2$ tells me that $(x+1)^4 + a^2$ is irreducible and hence $x^4+a^2$ is irreducible.



For $a$ even I have had no such luck. I know that the polynomial is not irreducible for all such $a$ because when $a = 2$, for example, we have the factorization $x^4+4 = (x^2-2x+2)(x^2+2x+2)$. It's easy to show that this polynomial has no linear factors, but I'm at a loss trying to decide when there are a pair of irreducible quadratic factors.



A thought I had was to try and consider when $mathbbQ(sqrtai)$ is a degree $4$ extension, but this didn't seem to help.










share|cite|improve this question









$endgroup$







  • 2




    $begingroup$
    If $a=2n^2$, then $x^4+a^2=x^4+(2n^2)^2=(x^2-2nx+2n^2)(x^2+2nx+2n^2)$. In other cases it seems to be irreducible.
    $endgroup$
    – Sil
    Mar 26 at 21:08














3












3








3


1



$begingroup$


For an integer $a$, I'm trying to find a criterion to tell me if $x^4+a^2$ is irreducible over $mathbbQ$.



What I've done so far is shown that if $a$ is odd, then $a^2$ is congruent to $1 mod 4$, and so the Eisenstein criterion with $p = 2$ tells me that $(x+1)^4 + a^2$ is irreducible and hence $x^4+a^2$ is irreducible.



For $a$ even I have had no such luck. I know that the polynomial is not irreducible for all such $a$ because when $a = 2$, for example, we have the factorization $x^4+4 = (x^2-2x+2)(x^2+2x+2)$. It's easy to show that this polynomial has no linear factors, but I'm at a loss trying to decide when there are a pair of irreducible quadratic factors.



A thought I had was to try and consider when $mathbbQ(sqrtai)$ is a degree $4$ extension, but this didn't seem to help.










share|cite|improve this question









$endgroup$




For an integer $a$, I'm trying to find a criterion to tell me if $x^4+a^2$ is irreducible over $mathbbQ$.



What I've done so far is shown that if $a$ is odd, then $a^2$ is congruent to $1 mod 4$, and so the Eisenstein criterion with $p = 2$ tells me that $(x+1)^4 + a^2$ is irreducible and hence $x^4+a^2$ is irreducible.



For $a$ even I have had no such luck. I know that the polynomial is not irreducible for all such $a$ because when $a = 2$, for example, we have the factorization $x^4+4 = (x^2-2x+2)(x^2+2x+2)$. It's easy to show that this polynomial has no linear factors, but I'm at a loss trying to decide when there are a pair of irreducible quadratic factors.



A thought I had was to try and consider when $mathbbQ(sqrtai)$ is a degree $4$ extension, but this didn't seem to help.







abstract-algebra field-theory irreducible-polynomials






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Mar 26 at 20:53









JonHalesJonHales

520311




520311







  • 2




    $begingroup$
    If $a=2n^2$, then $x^4+a^2=x^4+(2n^2)^2=(x^2-2nx+2n^2)(x^2+2nx+2n^2)$. In other cases it seems to be irreducible.
    $endgroup$
    – Sil
    Mar 26 at 21:08













  • 2




    $begingroup$
    If $a=2n^2$, then $x^4+a^2=x^4+(2n^2)^2=(x^2-2nx+2n^2)(x^2+2nx+2n^2)$. In other cases it seems to be irreducible.
    $endgroup$
    – Sil
    Mar 26 at 21:08








2




2




$begingroup$
If $a=2n^2$, then $x^4+a^2=x^4+(2n^2)^2=(x^2-2nx+2n^2)(x^2+2nx+2n^2)$. In other cases it seems to be irreducible.
$endgroup$
– Sil
Mar 26 at 21:08





$begingroup$
If $a=2n^2$, then $x^4+a^2=x^4+(2n^2)^2=(x^2-2nx+2n^2)(x^2+2nx+2n^2)$. In other cases it seems to be irreducible.
$endgroup$
– Sil
Mar 26 at 21:08











2 Answers
2






active

oldest

votes


















3












$begingroup$

Assuming, without loss of generality, $a>0$, the polynomial can be rewritten as
$$
x^4+2ax^2+a^2-2ax^2=(x^2+a)^2-(sqrt2ax)^2=
(x^2-sqrt2ax+a)(x^2+sqrt2ax+a)
$$

and it's obvious that the two polynomials are irreducible over $mathbbR$. By uniqueness of factorization, this is a factorization in $mathbbQ[x]$ if and only if $2a$ is a square in $mathbbQ$.






share|cite|improve this answer









$endgroup$




















    3












    $begingroup$

    Notice that we are trying to reduce that polynomial by this way:



    $$x^4+a^2=(x^2-bx+a)(x^2+bx+a)=x^4+(2a-b^2)x^2+a^2$$



    We need:



    $$2a-b^2=0$$
    $$b=sqrt2a$$



    But since we are working on integers then $$a=2k^2$$ .So your polynomial is reducible if and only if it can be written i nthis form:



    $$x^4+4k^4=(x^2-2kx+2k^2)(x^2+2kx+2k^2)$$



    Which is also known as Sophie Germain Identity.






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      I think you should justify why $(x^2-bx+a)(x^2+bx+a)$ is the only possible form, and not generic $(x^2+mx+n)(x^2+px+q)$
      $endgroup$
      – Sil
      Mar 26 at 21:27










    • $begingroup$
      @Sil It's pretty easy with complex factorization. And it can also be done with a system of equations.
      $endgroup$
      – Eureka
      Mar 26 at 21:30







    • 1




      $begingroup$
      @Sil Try expanding out $(x^2+mx+n)(x^2+px+q)$ and look at the $x^3$ term to why $m=-p$. Then expand $(x^2-bx+n)(x^2+bx+q)$ and look at the $x$ term to see why $n=q$.
      $endgroup$
      – Ethan MacBrough
      Mar 26 at 21:40










    • $begingroup$
      @EthanMacBrough I understand, my point though was that things like that should be in answer itself.
      $endgroup$
      – Sil
      Mar 26 at 21:42











    Your Answer








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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3












    $begingroup$

    Assuming, without loss of generality, $a>0$, the polynomial can be rewritten as
    $$
    x^4+2ax^2+a^2-2ax^2=(x^2+a)^2-(sqrt2ax)^2=
    (x^2-sqrt2ax+a)(x^2+sqrt2ax+a)
    $$

    and it's obvious that the two polynomials are irreducible over $mathbbR$. By uniqueness of factorization, this is a factorization in $mathbbQ[x]$ if and only if $2a$ is a square in $mathbbQ$.






    share|cite|improve this answer









    $endgroup$

















      3












      $begingroup$

      Assuming, without loss of generality, $a>0$, the polynomial can be rewritten as
      $$
      x^4+2ax^2+a^2-2ax^2=(x^2+a)^2-(sqrt2ax)^2=
      (x^2-sqrt2ax+a)(x^2+sqrt2ax+a)
      $$

      and it's obvious that the two polynomials are irreducible over $mathbbR$. By uniqueness of factorization, this is a factorization in $mathbbQ[x]$ if and only if $2a$ is a square in $mathbbQ$.






      share|cite|improve this answer









      $endgroup$















        3












        3








        3





        $begingroup$

        Assuming, without loss of generality, $a>0$, the polynomial can be rewritten as
        $$
        x^4+2ax^2+a^2-2ax^2=(x^2+a)^2-(sqrt2ax)^2=
        (x^2-sqrt2ax+a)(x^2+sqrt2ax+a)
        $$

        and it's obvious that the two polynomials are irreducible over $mathbbR$. By uniqueness of factorization, this is a factorization in $mathbbQ[x]$ if and only if $2a$ is a square in $mathbbQ$.






        share|cite|improve this answer









        $endgroup$



        Assuming, without loss of generality, $a>0$, the polynomial can be rewritten as
        $$
        x^4+2ax^2+a^2-2ax^2=(x^2+a)^2-(sqrt2ax)^2=
        (x^2-sqrt2ax+a)(x^2+sqrt2ax+a)
        $$

        and it's obvious that the two polynomials are irreducible over $mathbbR$. By uniqueness of factorization, this is a factorization in $mathbbQ[x]$ if and only if $2a$ is a square in $mathbbQ$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Mar 26 at 21:28









        egregegreg

        186k1486209




        186k1486209





















            3












            $begingroup$

            Notice that we are trying to reduce that polynomial by this way:



            $$x^4+a^2=(x^2-bx+a)(x^2+bx+a)=x^4+(2a-b^2)x^2+a^2$$



            We need:



            $$2a-b^2=0$$
            $$b=sqrt2a$$



            But since we are working on integers then $$a=2k^2$$ .So your polynomial is reducible if and only if it can be written i nthis form:



            $$x^4+4k^4=(x^2-2kx+2k^2)(x^2+2kx+2k^2)$$



            Which is also known as Sophie Germain Identity.






            share|cite|improve this answer









            $endgroup$












            • $begingroup$
              I think you should justify why $(x^2-bx+a)(x^2+bx+a)$ is the only possible form, and not generic $(x^2+mx+n)(x^2+px+q)$
              $endgroup$
              – Sil
              Mar 26 at 21:27










            • $begingroup$
              @Sil It's pretty easy with complex factorization. And it can also be done with a system of equations.
              $endgroup$
              – Eureka
              Mar 26 at 21:30







            • 1




              $begingroup$
              @Sil Try expanding out $(x^2+mx+n)(x^2+px+q)$ and look at the $x^3$ term to why $m=-p$. Then expand $(x^2-bx+n)(x^2+bx+q)$ and look at the $x$ term to see why $n=q$.
              $endgroup$
              – Ethan MacBrough
              Mar 26 at 21:40










            • $begingroup$
              @EthanMacBrough I understand, my point though was that things like that should be in answer itself.
              $endgroup$
              – Sil
              Mar 26 at 21:42















            3












            $begingroup$

            Notice that we are trying to reduce that polynomial by this way:



            $$x^4+a^2=(x^2-bx+a)(x^2+bx+a)=x^4+(2a-b^2)x^2+a^2$$



            We need:



            $$2a-b^2=0$$
            $$b=sqrt2a$$



            But since we are working on integers then $$a=2k^2$$ .So your polynomial is reducible if and only if it can be written i nthis form:



            $$x^4+4k^4=(x^2-2kx+2k^2)(x^2+2kx+2k^2)$$



            Which is also known as Sophie Germain Identity.






            share|cite|improve this answer









            $endgroup$












            • $begingroup$
              I think you should justify why $(x^2-bx+a)(x^2+bx+a)$ is the only possible form, and not generic $(x^2+mx+n)(x^2+px+q)$
              $endgroup$
              – Sil
              Mar 26 at 21:27










            • $begingroup$
              @Sil It's pretty easy with complex factorization. And it can also be done with a system of equations.
              $endgroup$
              – Eureka
              Mar 26 at 21:30







            • 1




              $begingroup$
              @Sil Try expanding out $(x^2+mx+n)(x^2+px+q)$ and look at the $x^3$ term to why $m=-p$. Then expand $(x^2-bx+n)(x^2+bx+q)$ and look at the $x$ term to see why $n=q$.
              $endgroup$
              – Ethan MacBrough
              Mar 26 at 21:40










            • $begingroup$
              @EthanMacBrough I understand, my point though was that things like that should be in answer itself.
              $endgroup$
              – Sil
              Mar 26 at 21:42













            3












            3








            3





            $begingroup$

            Notice that we are trying to reduce that polynomial by this way:



            $$x^4+a^2=(x^2-bx+a)(x^2+bx+a)=x^4+(2a-b^2)x^2+a^2$$



            We need:



            $$2a-b^2=0$$
            $$b=sqrt2a$$



            But since we are working on integers then $$a=2k^2$$ .So your polynomial is reducible if and only if it can be written i nthis form:



            $$x^4+4k^4=(x^2-2kx+2k^2)(x^2+2kx+2k^2)$$



            Which is also known as Sophie Germain Identity.






            share|cite|improve this answer









            $endgroup$



            Notice that we are trying to reduce that polynomial by this way:



            $$x^4+a^2=(x^2-bx+a)(x^2+bx+a)=x^4+(2a-b^2)x^2+a^2$$



            We need:



            $$2a-b^2=0$$
            $$b=sqrt2a$$



            But since we are working on integers then $$a=2k^2$$ .So your polynomial is reducible if and only if it can be written i nthis form:



            $$x^4+4k^4=(x^2-2kx+2k^2)(x^2+2kx+2k^2)$$



            Which is also known as Sophie Germain Identity.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Mar 26 at 21:23









            EurekaEureka

            1,189116




            1,189116











            • $begingroup$
              I think you should justify why $(x^2-bx+a)(x^2+bx+a)$ is the only possible form, and not generic $(x^2+mx+n)(x^2+px+q)$
              $endgroup$
              – Sil
              Mar 26 at 21:27










            • $begingroup$
              @Sil It's pretty easy with complex factorization. And it can also be done with a system of equations.
              $endgroup$
              – Eureka
              Mar 26 at 21:30







            • 1




              $begingroup$
              @Sil Try expanding out $(x^2+mx+n)(x^2+px+q)$ and look at the $x^3$ term to why $m=-p$. Then expand $(x^2-bx+n)(x^2+bx+q)$ and look at the $x$ term to see why $n=q$.
              $endgroup$
              – Ethan MacBrough
              Mar 26 at 21:40










            • $begingroup$
              @EthanMacBrough I understand, my point though was that things like that should be in answer itself.
              $endgroup$
              – Sil
              Mar 26 at 21:42
















            • $begingroup$
              I think you should justify why $(x^2-bx+a)(x^2+bx+a)$ is the only possible form, and not generic $(x^2+mx+n)(x^2+px+q)$
              $endgroup$
              – Sil
              Mar 26 at 21:27










            • $begingroup$
              @Sil It's pretty easy with complex factorization. And it can also be done with a system of equations.
              $endgroup$
              – Eureka
              Mar 26 at 21:30







            • 1




              $begingroup$
              @Sil Try expanding out $(x^2+mx+n)(x^2+px+q)$ and look at the $x^3$ term to why $m=-p$. Then expand $(x^2-bx+n)(x^2+bx+q)$ and look at the $x$ term to see why $n=q$.
              $endgroup$
              – Ethan MacBrough
              Mar 26 at 21:40










            • $begingroup$
              @EthanMacBrough I understand, my point though was that things like that should be in answer itself.
              $endgroup$
              – Sil
              Mar 26 at 21:42















            $begingroup$
            I think you should justify why $(x^2-bx+a)(x^2+bx+a)$ is the only possible form, and not generic $(x^2+mx+n)(x^2+px+q)$
            $endgroup$
            – Sil
            Mar 26 at 21:27




            $begingroup$
            I think you should justify why $(x^2-bx+a)(x^2+bx+a)$ is the only possible form, and not generic $(x^2+mx+n)(x^2+px+q)$
            $endgroup$
            – Sil
            Mar 26 at 21:27












            $begingroup$
            @Sil It's pretty easy with complex factorization. And it can also be done with a system of equations.
            $endgroup$
            – Eureka
            Mar 26 at 21:30





            $begingroup$
            @Sil It's pretty easy with complex factorization. And it can also be done with a system of equations.
            $endgroup$
            – Eureka
            Mar 26 at 21:30





            1




            1




            $begingroup$
            @Sil Try expanding out $(x^2+mx+n)(x^2+px+q)$ and look at the $x^3$ term to why $m=-p$. Then expand $(x^2-bx+n)(x^2+bx+q)$ and look at the $x$ term to see why $n=q$.
            $endgroup$
            – Ethan MacBrough
            Mar 26 at 21:40




            $begingroup$
            @Sil Try expanding out $(x^2+mx+n)(x^2+px+q)$ and look at the $x^3$ term to why $m=-p$. Then expand $(x^2-bx+n)(x^2+bx+q)$ and look at the $x$ term to see why $n=q$.
            $endgroup$
            – Ethan MacBrough
            Mar 26 at 21:40












            $begingroup$
            @EthanMacBrough I understand, my point though was that things like that should be in answer itself.
            $endgroup$
            – Sil
            Mar 26 at 21:42




            $begingroup$
            @EthanMacBrough I understand, my point though was that things like that should be in answer itself.
            $endgroup$
            – Sil
            Mar 26 at 21:42

















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