Factorization of Polynomials Over a Field - Nicholson Example 10, Page 215

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SUMMARY

The discussion centers on Example 10 from W. Keith Nicholson's "Introduction to Abstract Algebra (Third Edition)," specifically regarding the factorization of polynomials over a field. The polynomial $$\overline{f(x)} = x^4 + x + 1$$ in $$\mathbb{Z}_2[x]$$ is analyzed, revealing that it has no roots in $$\mathbb{Z}_2$$, which leads to the conclusion that it may fail to be irreducible. A clarification was provided that the statement about irreducibility is conditional, emphasizing the importance of careful reading in mathematical contexts.

PREREQUISITES
  • Understanding of polynomial factorization in abstract algebra
  • Familiarity with finite fields, specifically $$\mathbb{Z}_2$$
  • Knowledge of irreducibility criteria for polynomials
  • Basic skills in reading and interpreting mathematical texts
NEXT STEPS
  • Study the concept of irreducibility in polynomial rings over fields
  • Explore examples of polynomial factorization in $$\mathbb{Z}_p[x]$$ for various primes $$p$$
  • Learn about the implications of having roots in finite fields
  • Review Section 4.2 of Nicholson's book for additional context and examples
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Students of abstract algebra, mathematicians focusing on polynomial theory, and educators teaching concepts related to factorization over fields.

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I am reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) ...

I am focused on Section 4.2:Factorization of Polynomials over a Filed.

I need some help with Example 10 on page 215 ...

The relevant text from Nicholson's book is as follows:View attachment 4591In the above text, we read the following:

" ... ... Reduction modulo $$2$$ gives $$\overline{f(x)} = x^4 + x + 1$$ in $$\mathbb{Z}_2 [x]$$. This polynomial has no roots in $$\mathbb{Z}_2$$, so it fails to be irreducible ... ... "Can someone please explain the reasoning behind the statement that since the polynomial has no roots in $$\mathbb{Z}_2$$ then it fails to be irreducible?

Hope someone can help ... ...

Peter
 
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Hi Peter,

You have missed an IF it fails to be irreducible...
 
Fallen Angel said:
Hi Peter,

You have missed an IF it fails to be irreducible...
Thanks Fallen Angel ... you are quite right ...

hmm ... must read more carefully ... very careless of me :(

Thanks again for your help ...

Peter
 

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