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Chapter 17: Problem 3

Determine all polynomials \(f(x) \in \mathbf{Z}_{2}[x]\) such that \(1 \leq\) degree \(f(x) \leq 3\) and \(f(x)\) is irreducible (over \(\mathbf{Z}_{2}\) ).

Short Answer

Expert verified
The irreducible polynomials of degree 1 to 3 over \(\mathbf{Z}_{2}\) are \(x\), \(x+1\), \(x^{2}+1\), \(x^{2}+x+1\), \(x^{3}+1\), \(x^{3}+x+1\), \(x^{3}+x^{2}+1\), \(x^{3}+x^{2}+x+1\).

Step by step solution

01

List all polynomials

The field \(\mathbf{Z}_{2}\) contains only two elements, 0 and 1. So, one can list all the polynomials of degree 1 to 3. The polynomials of degree 1 are \(f(x)=x\), \(f(x)=x+1\). The polynomials of degree 2 are \(f(x)=x^{2}\), \(f(x)=x^{2}+1\), \(f(x)=x^{2}+x\), \(f(x)=x^{2}+x+1\). The polynomials of degree 3 are \(f(x)=x^{3}\), \(f(x)=x^{3}+1\), \(f(x)=x^{3}+x\), \(f(x)=x^{3}+x+1\), \(f(x)=x^{3}+x^{2}\), \(f(x)=x^{3}+x^{2}+1\), \(f(x)=x^{3}+x^{2}+x\), \(f(x)=x^{3}+x^{2}+x+1\).
02

Check for reducibility

Next, we can use the definition of reducibility to eliminate those polynomials that can be factored into the product of two non-constant polynomials. In \(\mathbf{Z}_{2}[x]\), all polynomials of degree 1 are irreducible. For degree 2, we have to exclude \(x^{2}\) and \(x^{2}+x\), because they can be written as \(x \cdot x\) and \(x \cdot (x+1)\), respectively. In the case of degree 3, we exclude \(x^{3}\) (\(= x \cdot x^{2}\)), \(x^{3} + x^{2}\) (\(= x^{2} \cdot (x+1)\)), \(x^{3}+x\) (\(= x \cdot (x^{2} + 1)\)), and \(x^{3}+x^{2}+x\) (\(= x \cdot (x^{2} + x + 1)\)).
03

List remaining polynomials

Now, list all remaining polynomials: polynomials of degree 1: \(x\), \(x+1\), polynomials of degree 2: \(x^{2}+1\), \(x^{2}+x+1\), polynomials of degree 3: \(x^{3}+1\), \(x^{3}+x+1\), \(x^{3}+x^{2}+1\), \(x^{3}+x^{2}+x+1\).

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Most popular questions from this chapter

a) Find all roots of \(f(x)=x^{2}+4 x\) if \(f(x) \in \mathbf{Z}_{12}[x]\). b) Find four distinct linear polynomials \(g(x), h(x), s(x), t(x) \in \mathbf{Z}_{12}[x]\) so that \(f(x)=\) \(g(x) h(x)=s(x) r(x)\) c) Do the results in part (b) contradict the statements made in the paragraph following Example 17.7?

\begin{aligned} &\text { Let } f(x), g(x) \in Z_{7}[x] \text { where } f(x)=2 x^{4}+2 x^{3}+3 x^{2}+x+4 \text { and } g(x)=3 x^{3}+5 x^{2}+ \\ &6 x+1 \text {. Determine } f(x)+g(x), f(x)-g(x), \text { and } f(x) g(x) \end{aligned}

Determine whether or not each of the following polynomials is irreducible over the given fields. If it is reducible, provide a factorization into irreducible factors. a) \(x^{2}+3 x-1\) over \(Q, R, C\) b) \(x^{4}-2\) over \(\mathbf{Q}, \mathbf{R}, \mathbf{C}\) c) \(x^{2}+x+1\) over \(\mathbf{Z}_{3}, \mathbf{Z}_{5}, \mathbf{Z}\) ? d) \(x^{4}+x^{3}+1\) over \(\mathbf{Z}_{2}\) e) \(x^{3}+x+1\) over \(Z_{5}\) f) \(x^{3}+3 x^{2}-x+1\) over \(\mathbf{Z}_{5}\)

If \(R\) is an integral domain, prove that if \(f(x)\) is a unit in \(R[x]\), then \(f(x)\) is a constant and is a unit in \(R\).

Give the characteristic for each of the following rings: a) \(\mathbf{Z}_{11}\) b) Z.s. \([x]\) c) \(Q[x]\) d) \(\mathbf{Z}[\sqrt{5}]=\\{a+b \sqrt{5} \mid a, b \in \mathbf{Z}\\}\), under the ordinary operations of addition and multiplication of real numbers
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