Advanced Engineering Mathematics
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ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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#7
Please include reasons for each step. Thanks!
![**Problem 7:** Let \( \alpha \) be a zero of \( f(x) = x^2 + 2x + 2 \) in some extension field of \( \mathbb{Z}_3 \). Find the other zero of \( f(x) \) in \( \mathbb{Z}_3[\alpha] \).
### Explanation
In this mathematical problem, we are asked to find the roots of the polynomial \( f(x) = x^2 + 2x + 2 \) over the extension field \( \mathbb{Z}_3[\alpha] \). Here, \( \alpha \) is a zero of \( f(x) \), meaning \( f(\alpha) = 0 \).
**Steps to Solve:**
1. **Identify the given polynomial:**
- \( f(x) = x^2 + 2x + 2 \)
2. **Field Extension:**
- We are working within \( \mathbb{Z}_3 \), the finite field with three elements, extended by \( \alpha \).
3. **Objective:**
- Find the other zero besides \( \alpha \).
### Further Explanation
- **Polynomial Roots:** In any extension field, a quadratic polynomial has two roots (counting multiplicities). Here, we require finding the second zero in the given extension field.
- **Extension Fields:** This involves constructing a field where the original polynomial splits completely if it does not in the base field.
### Example Approach:
- Use the factored form:
\[
f(x) = (x - \alpha)(x - \beta)
\]
Here, \( \beta \) is the root you need to find.
- Consider known identities and the nature of finite fields to determine \( \beta \).
This type of problem is common in abstract algebra and Galois theory, where finite fields and root-finding of polynomials play a significant role.](https://content.bartleby.com/qna-images/question/174d7d89-86f6-4fd3-817b-704de3f40cca/f5e19135-5b32-408f-9d80-6e8cf04e2795/8cy0qov_processed.png)
Transcribed Image Text:**Problem 7:** Let \( \alpha \) be a zero of \( f(x) = x^2 + 2x + 2 \) in some extension field of \( \mathbb{Z}_3 \). Find the other zero of \( f(x) \) in \( \mathbb{Z}_3[\alpha] \).
### Explanation
In this mathematical problem, we are asked to find the roots of the polynomial \( f(x) = x^2 + 2x + 2 \) over the extension field \( \mathbb{Z}_3[\alpha] \). Here, \( \alpha \) is a zero of \( f(x) \), meaning \( f(\alpha) = 0 \).
**Steps to Solve:**
1. **Identify the given polynomial:**
- \( f(x) = x^2 + 2x + 2 \)
2. **Field Extension:**
- We are working within \( \mathbb{Z}_3 \), the finite field with three elements, extended by \( \alpha \).
3. **Objective:**
- Find the other zero besides \( \alpha \).
### Further Explanation
- **Polynomial Roots:** In any extension field, a quadratic polynomial has two roots (counting multiplicities). Here, we require finding the second zero in the given extension field.
- **Extension Fields:** This involves constructing a field where the original polynomial splits completely if it does not in the base field.
### Example Approach:
- Use the factored form:
\[
f(x) = (x - \alpha)(x - \beta)
\]
Here, \( \beta \) is the root you need to find.
- Consider known identities and the nature of finite fields to determine \( \beta \).
This type of problem is common in abstract algebra and Galois theory, where finite fields and root-finding of polynomials play a significant role.
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