Exploring Irreducible Polynomial and Reducing Techniques for g(x)

  • Thread starter ElDavidas
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In summary, The polynomial g(x) = x^8+ x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 is not irreducible as it can be factored into (1+x+x^2)(x^6+x^3+1). To further reduce the polynomial, you can use the roots of g(x)(x-1) = (x^9-1) which are \alpha = cos(120)+isin(120)=w and \alpha = cos(240)+isin(240)=w^2. Combining these two roots, you can get (1+x+x^2)(x^6+x^3+1
  • #1
ElDavidas
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Hi, I'm trying to show whether the polynomial

[tex]g(x) = x^8+ x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 [/tex]

is irreducible or not.

So far I have evaluated [itex] g(x+1) [/itex] and applied Eisenstein's theorem to it. From what I gather it doesn't appear to be irreducible. Is this right, because I reckon it should be irreducible? This may just be a simple calculation error.

And if g(x) is reducible, how do I go about reducing the polynomial more?

Thanks
 
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  • #2
What is g(x)(x-1), and what are its roots?
 
  • #3
Of course, that would be helpful.

[tex]g(x)(x-1) = (x^9-1)[/tex]

and the roots are [itex] \alpha [/itex] existing in the complex numbers such that
[itex] \alpha^9 = 1 [/itex]
 
Last edited:
  • #4
So you can solve your problem now right?
 
  • #5
Two of the roots are cos(120)+isin(120)=w (cube root of 1,) and cos(240)+isin(240)=w^2. Combining these two roots (x-w)(x-w^2)=x^2+x+1.

This then divides the polynominal giving: (1+x+x^2)(x^6+x^3+1).
 

FAQ: Exploring Irreducible Polynomial and Reducing Techniques for g(x)

What is an irreducible polynomial?

An irreducible polynomial is a polynomial that cannot be factored into smaller polynomials with coefficients in the same field. In other words, it has no non-trivial factors.

How do I determine if a polynomial is irreducible?

One way to determine if a polynomial is irreducible is to use the Eisenstein's criterion. If the polynomial satisfies Eisenstein's criterion with respect to a prime number, then it is irreducible.

What is the importance of irreducible polynomials?

Irreducible polynomials play a crucial role in algebra, number theory, and cryptography. They are used to construct finite fields, which have applications in coding theory, error-correcting codes, and encryption algorithms.

Can irreducible polynomials have complex coefficients?

Yes, irreducible polynomials can have complex coefficients. In fact, any polynomial with complex coefficients can be reduced to a product of irreducible polynomials, similar to the fundamental theorem of algebra.

Are all polynomials irreducible?

No, not all polynomials are irreducible. In fact, most polynomials can be factored into irreducible polynomials. Only a small subset of polynomials, such as prime polynomials or cyclotomic polynomials, are irreducible.

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