Exploring Irreducible Polynomial and Reducing Techniques for g(x)

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Discussion Overview

The discussion revolves around the irreducibility of the polynomial g(x) = x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1. Participants explore techniques for determining irreducibility, including evaluations of g(x+1) and the application of Eisenstein's theorem, as well as methods for reducing the polynomial if it is found to be reducible.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant questions the irreducibility of g(x) after applying Eisenstein's theorem and evaluating g(x+1), suggesting a possible calculation error.
  • Another participant introduces the product g(x)(x-1) and identifies it as (x^9 - 1), prompting a discussion about its roots.
  • A participant notes that the roots of g(x)(x-1) are the ninth roots of unity, specifically mentioning two roots as cos(120) + i sin(120) and cos(240) + i sin(240), which correspond to cube roots of unity.
  • It is proposed that these roots can be combined to form a factorization of g(x), leading to the expression (1 + x + x^2)(x^6 + x^3 + 1).

Areas of Agreement / Disagreement

Participants express differing views on the irreducibility of g(x), with some suggesting it may be reducible based on their calculations, while others provide a factorization that implies it is reducible. The discussion remains unresolved regarding the status of g(x) as irreducible or reducible.

Contextual Notes

Participants have not reached a consensus on the application of Eisenstein's theorem or the implications of the polynomial's factorization. There are also unresolved aspects regarding the completeness of the reduction process and the assumptions made during the evaluation of g(x+1).

ElDavidas
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Hi, I'm trying to show whether the polynomial

g(x) = x^8+ x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1

is irreducible or not.

So far I have evaluated g(x+1) 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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What is g(x)(x-1), and what are its roots?
 
Of course, that would be helpful.

g(x)(x-1) = (x^9-1)

and the roots are \alpha existing in the complex numbers such that
\alpha^9 = 1
 
Last edited:
So you can solve your problem now right?
 
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).
 

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