Is x^6-2x^3-1 Irreducible Over Q?

  • Thread starter T-O7
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In summary, the conversation discusses whether the polynomial x^6-2x^3-1 is irreducible over Q. It is determined that Eisenstein's criterion does not work and reducing modulo 2 and 3 leads to a potentially irreducible polynomial. The conversation also explores methods for determining irreducibility over F3. It is ultimately concluded that the original polynomial is irreducible over Q.
  • #1
T-O7
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Hi,
I need to figure out whether or not the polynomial
[tex]x^6-2x^3-1[/tex]
is irreducible (over Q).
I don't think Eisenstein works in this case, and performing modulo 2 on this i get [tex]x^6-1[/tex]
which is reducible over F2.
Any ideas? Incidently, if i let y=x^3, then i get
[tex]y^2-2y -1[/tex]
which is irreducible over Q...but I'm not sure if that means anything about the original polynomial.
 
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  • #2
Sure enough, Eisenstein doesn't work as there is no prime [tex] p[/tex] such that [tex] p\mid -1[/tex]. However if you reduce modulo 3, you get (i hope)
[tex] x^6-[2]x^3-[1] = x^6+[-2]x^3+[-1] = x^6+x^3+[2][/tex] and this polynomial in [tex] Z_2[x][/tex] seems to be irreducible (at a first try I couldn't factor it in 2 polynomials of smaller degree, but you can try yourself). So if that polynomial is irreducible in [tex] Z_2[x][/tex] then [tex]x^6-2x^3-1[/tex] is irreducible in [tex] Q[x][/tex].
 
  • #3
Hmm..okay. I was hoping I wouldn't have to resort to brute force hehe
OK, so I have shown that the polynomial [tex] x^6+x^3+2[/tex] has no linear or quadratic factors over F3, but how did you show it can't factor into cubic terms? I don't really know the irreducible cubic polynomials over F3, and it seems like there might be quite a few of them?
 
  • #4
T-O7 said:
I don't really know the irreducible cubic polynomials over F3, and it seems like there might be quite a few of them?

It won't be horrid, there are only 18 candidates for monic irreducibles of degree 3 (constant term must be non-zero), 10 of them will have roots. Trial division by the remaining 8 will be tedious though. Actually you can cut this down quite alot, if you check that none of the monic irreducibles with constant term 2 divide your polynomial, then you know none with constant term 1 do (do you see why?).

Sorry, I don't have anything quicker to offer.
 
  • #5
Yes, great, thanks for the tip. After a little tedious work, it turns out that it is irreducible over F3, so my original polynomial was irreducible over Q. Yay, thanks a lot :biggrin:
 

1. What does it mean for a polynomial to be irreducible over Q?

Irreducibility over Q means that the polynomial cannot be factored into smaller polynomials with coefficients in the rational numbers, Q.

2. Why is it important to determine if a polynomial is irreducible over Q?

Determining irreducibility over Q helps us understand the structure of the polynomial and its roots. It also has applications in fields such as algebraic number theory and cryptography.

3. How can I determine if x^6-2x^3-1 is irreducible over Q?

One method is to use the Rational Root Theorem to check for any possible rational roots. If no rational roots are found, then a more advanced method, such as Eisenstein's criterion, can be used to determine irreducibility.

4. What is Eisenstein's criterion and how does it apply to x^6-2x^3-1?

Eisenstein's criterion states that if a polynomial has integer coefficients and there exists a prime number p that divides all coefficients except the leading coefficient, and p^2 does not divide the constant term, then the polynomial is irreducible over Q. In the case of x^6-2x^3-1, the prime number 3 divides all coefficients except the leading coefficient and 3^2 does not divide the constant term (-1), so the polynomial is irreducible over Q.

5. Is there a general method for determining the irreducibility of polynomials over Q?

No, there is no one method that can determine the irreducibility of all polynomials over Q. However, there are several criteria, such as Eisenstein's criterion, that can be useful in determining irreducibility for specific polynomials.

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