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

In summary, Peter is reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) and is focused on Section 4.2:Factorization of Polynomials over a Field. He needs help with Example 10 on page 215 and is asking for an explanation behind the statement that since the polynomial has no roots in \mathbb{Z}_2 then it fails to be irreducible. Peter thanks Fallen Angel for pointing out his mistake and acknowledges that he needs to read more carefully.
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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 \(\displaystyle 2\) gives \(\displaystyle \overline{f(x)} = x^4 + x + 1\) in \(\displaystyle \mathbb{Z}_2 [x]\). This polynomial has no roots in \(\displaystyle \mathbb{Z}_2\), so it fails to be irreducible ... ... "Can someone please explain the reasoning behind the statement that since the polynomial has no roots in \(\displaystyle \mathbb{Z}_2\) then it fails to be irreducible?

Hope someone can help ... ...

Peter
 
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  • #2
Hi Peter,

You have missed an IF it fails to be irreducible...
 
  • #3
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
 

1. What is the purpose of factorization of polynomials over a field?

The purpose of factorization of polynomials over a field is to break down a polynomial into smaller, simpler factors. This can help in solving equations, finding roots, and understanding the behavior of the polynomial.

2. What is a field in the context of polynomial factorization?

In mathematics, a field is a set of numbers that is closed under addition, subtraction, multiplication, and division. In the context of polynomial factorization, a field is a set of numbers that contains the coefficients and variables of the polynomial being factored.

3. How do you perform factorization of polynomials over a field?

To factorize a polynomial over a field, you can use methods such as grouping, factoring by grouping, or using the rational root theorem. These methods involve finding common factors, grouping terms, and using the properties of the field to simplify the polynomial.

4. Why is it important to factorize polynomials over a field?

Factoring polynomials over a field is important because it allows us to find the roots of the polynomial, which are the values of the variable that make the polynomial equal to zero. It also helps in simplifying and solving equations involving polynomials, and understanding the behavior of the polynomial.

5. Can all polynomials be factorized over a field?

No, not all polynomials can be factorized over a field. The ability to factorize a polynomial over a field depends on the properties of the field and the polynomial itself. For example, a polynomial with irrational coefficients cannot be factorized over a field of rational numbers.

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