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

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The discussion centers on Example 10 from W. Keith Nicholson's "Introduction to Abstract Algebra," specifically regarding the factorization of the polynomial \( f(x) = x^4 + x + 1 \) over the field \( \mathbb{Z}_2 \). The polynomial is noted to have no roots in \( \mathbb{Z}_2 \), which leads to the conclusion that it may not be irreducible. However, a clarification is made that the statement about irreducibility is conditional, emphasizing the need for careful reading. Participants acknowledge the importance of understanding the implications of polynomial roots in relation to irreducibility. The conversation highlights the nuances of polynomial factorization in abstract algebra.
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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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