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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
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