micromass said:How would you approach this question in [tex]\mathbb{F}_2[X][/tex]?? Also think about which parts of this approach do and do not carry over to [tex]\mathbb{F}_8[X][/tex]??
R.P.F. said:[tex]q=p^r[/tex]There is a theorem saying that the irreducible polynomials of [tex]x^q-x[/tex] over [tex]F_p[x][/tex] where [tex]p[/tex] is a prime are the irreducible polynomials in [tex]F_p[x][/tex] whose degrees divide [tex]r[/tex]. So in [tex]F_2[x][/tex] the poly is [tex]x(x-1)(x^2+x+1)(x^4+x^3+x^3+x+1)(x^4+x+1)(x^4+x^3+1)[/tex]. I can show that in [tex]F_8[x][/tex] the poly has no more linear factors than in [tex]F_2[/tex] but not sure how to deal with quadratics.
micromass said:OK, that's already good. Can you show now that the expression [tex]x(x-1)(x^2+x+1)(x^4+x^3+x^3+x+1)(x^4+x+1)(x^4+x^3+1)[/tex] is also the factorization in [tex]\mathbb{F}_8[X][/tex]. To do this, you have to show that no other polynomial in the expression factorizes. Let me give you an example how to do this:
Consider [tex]x^2+x+1[/tex], if it factorizes in [tex]\mathbb{F}_8[X][/tex], then it factorizes in linear factors. So, if it factorizes, then the polynomial [tex]x^2+x+1[/tex] must have a root in [tex]\mathbb{F}_8[/tex]. This is not possible, because no element in [tex]\mathbb{F}_8[/tex] has degree 2.
Try to do something for the other polynomials...
I do not see how to see the degree-4 factors cannot be factored into two quadratics
Factoring polynomials is the process of breaking down a polynomial into smaller, simpler polynomials. It involves finding the common factors of the polynomial and rewriting it as a product of these factors.
Factoring polynomials is important because it allows us to solve equations and find the roots of polynomials. It also helps us simplify complex expressions and make calculations easier.
The most commonly used methods for factoring polynomials are the Greatest Common Factor (GCF) method, the Difference of Squares method, and the Quadratic Formula method. Other methods include grouping, trial and error, and the sum and product rule.
No, not all polynomials can be factored. Some polynomials, such as prime polynomials, cannot be factored any further. However, most polynomials can be factored using one of the methods mentioned above.
To check if a polynomial is factored correctly, you can use the distributive property to multiply the factors back together and see if it equals the original polynomial. You can also use a graphing calculator to graph the polynomial and see if the factors match the x-intercepts.