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Chapter 17: Problem 18

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

Short Answer

Expert verified
The characteristics for the given rings are: a) \(\mathbf{Z}_{11}\) is 11; b) Z.s. \([x]\) is 0; c) \(Q[x]\) is 0; d) \(\mathbf{Z}[\sqrt{5}]\) is 0.

Step by step solution

01

Determine the characteristic of \(\mathbf{Z}_{11}\)

The ring \(\mathbf{Z}_{11}\) represents the set of integers modulo 11. Here, the characteristic is 11 because 11鈥�1 mod 11 is 0.
02

Determine the characteristic of Z.s. \([x]\)

The ring of integer polynomials is denoted by Z.s. \([x]\). Here, the characteristic is 0 because the only way to get zero through sum of units (which is 1) is to sum zero times.
03

Determine the characteristic of \(Q[x]\)

The ring \(Q[x]\) represents the set of all polynomials with rational coefficients. Here, the characteristic is also 0, for the same reason as in Step 2.
04

Determine the characteristic of \(\mathbf{Z}[\sqrt{5}]\)

Finally, the ring \(\mathbf{Z}[\sqrt{5}]\) represents the set of all real numbers of the form \(a+b \sqrt{5}\) where \(a\) and \(b\) are integers. Since the integers are a subset, the characteristic is 0, same as for the ring of integers.

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Most popular questions from this chapter

Let \(f(x), g(x) \in \mathbf{R}[x]\) with \(f(x)=x^{3}+2 x^{2}+a x-b, g(x)=x^{3}+x^{2}-b x+a\). Determine values for \(a, b\) so that the gcd of \(f(x), g(x)\) is a polynomial of degree 2 .

\text { Construct a finite field of } 27 \text { elements. }

\text { Determine all of the polynomials of degree } 2 \text { in } \mathbf{Z}_{2}[x] \text {. }

For each of the following pairs \(f(x), g(x)\), find \(q(x), r(x)\) so that \(g(x)=q(x) f(x)+r(x)\), where \(r(x)=0\) or degree \(r(x)<\) degree \(f(x)\). a) \(f(x), g(x) \in Q[x], f(x)=x^{4}-5 x^{3}+7 x, g(x)=x^{5}-2 x^{2}+5 x-3\) b) \(f(x), g(x) \in Z_{2}[x], f(x)=x^{2}+1, g(x)=x^{4}+x^{3}+x^{2}+x+1\) c) \(f(x), g(x) \in Z_{5}[x], f(x)=x^{2}+3 x+1, g(x)=x^{4}+2 x^{3}+x+4\)

a) If \(f(x)=x^{4}-16\), find its roots and factorization in \(Q[x]\). b) Answer part (a) for \(f(x) \in \mathbf{R}[x]\). d) Answer parts (a), (b), and (c) for \(f(x)=x^{4}-25\). c) Answer part (a) for \(f(x) \in \mathbf{C}[x]\).
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