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Chapter 17: Problem 3

How many polynomials are there of degree 2 in \(\mathbf{Z}_{11}[x]\) ? How many have degree 3 ? degree 4 ? degree \(n\), for \(n \in \mathbf{N}\) ?

Short Answer

Expert verified
There are 1331 polynomials of degree 2, 14641 polynomials of degree 3, 161051 polynomials of degree 4, and \(11^{n+1}\) polynomials of degree \(n\), for \(n \in \mathbf{N}\).

Step by step solution

01

Understanding the structure of a polynomial

A polynomial of degree \(n\) can be written in the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x^1 + a_0x^0\). There are \(n+1\) different \(a_i\) coefficients, each of which can be a number from 0 to 10 (i.e., may take any of the 11 different values in \(\mathbf{Z}_{11}\)).
02

Calculating the number of polynomials

Because each of the \(n+1\) terms can be chosen independently, and because each term has 11 possibilities, the total number of polynomials of degree \(n\) available is \(11^{n+1}\).
03

Giving the results for specific degrees

For degree 2, there are \(11^{2+1} = 1331\) polynomials. For degree 3, there are \(11^{3+1} = 14641\) polynomials, and for degree 4, there are \(11^{4+1} = 161051\) polynomials. For arbitrary degree \(n\), there are \(11^{n+1}\) polynomials.

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Most popular questions from this chapter

Determine the values of \(v, b, r, k\), and \(\lambda\) for the balanced incomplete block design associated with the projective plane that arises from \(A P(F)\) for the following choices of \(F:\) (a) \(\mathbf{Z}_{5}\) (b) \(\mathbf{Z}_{7}\) (c) \(G F(8)\).

If \(F\) is any field, let \(f(x), g(x) \in F[x]\). If \(f(x), g(x)\) are relatively prime, prove that there is no element \(a \in F\) with \(f(a)=0\) and \(g(a)=0\).

Find three \(7 \times 7\) Latin squares that are orthogonal in pairs. Rewrite these results in standard form.

Construct the affine plane \(A P\left(\mathbf{Z}_{3}\right)\). Determine its parallel classes and the corresponding Latin squares for the classes of finite nonzero slope.

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) \(f(x), g(x) \in \mathbf{Q}[x], \quad f(x)=x^{4}-5 x^{3}+7 x, \quad g(x)=\) \(x^{5}-2 x^{2}+5 x-3\) b) \(f(x), g(x) \in \mathbf{Z}_{2}[x], f(x)=x^{2}+1, g(x)=x^{4}+x^{3}+\) \(x^{2}+x+1\) c) \(f(x), g(x) \in \mathbf{Z}_{5}[x], f(x)=x^{2}+3 x+1, g(x)=x^{4}+\) \(2 x^{3}+x+4\)
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