10.6. Show that 2x + 1 is a unit in Z4[x]. Then, for any prime p, find a unit in Z,²[x] that is not a constant polynomial. 'p²
10.6. Show that 2x + 1 is a unit in Z4[x]. Then, for any prime p, find a unit in Z,²[x] that is not a constant polynomial. 'p²
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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Could you explain how to show 10.6 in great detail? I also included a list of theorems and definitions in my textbook as a reference. Really appreciate your help!
![**Definition 10.1.** Let \( R \) be a ring. A *polynomial* with coefficients in \( R \) is a formal expression
\[ a_0 + a_1x + a_2x^2 + \cdots + a_nx^n, \]
where \( a_i \in R \) and \( n \) is a nonnegative integer. Suppose that \( b_0 + b_1x + \cdots + b_mx^m \) is also a polynomial with coefficients in \( R \). Without loss of generality, let us say that \( n \leq m \). Then these polynomials are equal if and only if \( a_i = b_i \) for all \( i \leq n \) and \( b_i = 0 \) for all \( i > n \). The set of all polynomials with coefficients in \( R \) is denoted \( R[x] \).
**Example 10.1.** Let \( R = \mathbb{Z}_5 \). Then (inserting congruence class brackets for clarity), an example of a polynomial in \( R[x] \) would be
\[ f(x) = [3] + [2]x + [4]x^2. \]
As part of the above definition, we observe that \( f(x) = g(x) \), where \( g(x) = [3] + [2]x + [4]x^3 \).
**Definition 10.2.** Let \( R \) be a ring and let \( f(x) = a_0 + a_1x + \cdots + a_nx^n \in R[x] \). Further suppose that \( a_m \neq 0 \) but \( a_k = 0 \) for all \( k > m \). Then the *degree* of \( f(x) \) is \( m \), and we write \(\deg(f(x)) = m \). The *leading term* of \( f(x) \) is \( a_m x^m \), and the *leading coefficient* is \( a_m \). Note that the *zero polynomial*, 0, has no degree, leading term, or leading coefficient. A *constant polynomial* has degree 0 (or is the zero polynomial). If \( R \) has an](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2Fa8232f65-b4b2-435c-8965-ae2b705d3652%2Fapj3ms_processed.png&w=3840&q=75)
Transcribed Image Text:**Definition 10.1.** Let \( R \) be a ring. A *polynomial* with coefficients in \( R \) is a formal expression
\[ a_0 + a_1x + a_2x^2 + \cdots + a_nx^n, \]
where \( a_i \in R \) and \( n \) is a nonnegative integer. Suppose that \( b_0 + b_1x + \cdots + b_mx^m \) is also a polynomial with coefficients in \( R \). Without loss of generality, let us say that \( n \leq m \). Then these polynomials are equal if and only if \( a_i = b_i \) for all \( i \leq n \) and \( b_i = 0 \) for all \( i > n \). The set of all polynomials with coefficients in \( R \) is denoted \( R[x] \).
**Example 10.1.** Let \( R = \mathbb{Z}_5 \). Then (inserting congruence class brackets for clarity), an example of a polynomial in \( R[x] \) would be
\[ f(x) = [3] + [2]x + [4]x^2. \]
As part of the above definition, we observe that \( f(x) = g(x) \), where \( g(x) = [3] + [2]x + [4]x^3 \).
**Definition 10.2.** Let \( R \) be a ring and let \( f(x) = a_0 + a_1x + \cdots + a_nx^n \in R[x] \). Further suppose that \( a_m \neq 0 \) but \( a_k = 0 \) for all \( k > m \). Then the *degree* of \( f(x) \) is \( m \), and we write \(\deg(f(x)) = m \). The *leading term* of \( f(x) \) is \( a_m x^m \), and the *leading coefficient* is \( a_m \). Note that the *zero polynomial*, 0, has no degree, leading term, or leading coefficient. A *constant polynomial* has degree 0 (or is the zero polynomial). If \( R \) has an
![**10.6.** Show that \(2x + 1\) is a unit in \(\mathbb{Z}_4[x]\). Then, for any prime \(p\), find a unit in \(\mathbb{Z}_{p^2}[x]\) that is not a constant polynomial.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2Fa8232f65-b4b2-435c-8965-ae2b705d3652%2Fhlo9r9_processed.png&w=3840&q=75)
Transcribed Image Text:**10.6.** Show that \(2x + 1\) is a unit in \(\mathbb{Z}_4[x]\). Then, for any prime \(p\), find a unit in \(\mathbb{Z}_{p^2}[x]\) that is not a constant polynomial.
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