1.31. Let p be a prime and let q be a prime that divides p - 1. (a) Let a € F and let b = a(P-¹)/4. Prove that either b = 1 or else b has order q. (Recall that the order of b is the smallest k ≥ 1 such that bk = 1 in F. Hint. Use Proposition 1.30.) (b) Suppose that we want to find an element of F of order q. Using (a), we can randomly choose a value of a € F and check whether b = a(p-¹)/9 satisfies b 1. How likely are we to succeed? In other words, compute the value of the ratio 1} (Hint. Use Theorem 1.31.) #{a € F*: a(P-1)/q; Fa #FP
1.31. Let p be a prime and let q be a prime that divides p - 1. (a) Let a € F and let b = a(P-¹)/4. Prove that either b = 1 or else b has order q. (Recall that the order of b is the smallest k ≥ 1 such that bk = 1 in F. Hint. Use Proposition 1.30.) (b) Suppose that we want to find an element of F of order q. Using (a), we can randomly choose a value of a € F and check whether b = a(p-¹)/9 satisfies b 1. How likely are we to succeed? In other words, compute the value of the ratio 1} (Hint. Use Theorem 1.31.) #{a € F*: a(P-1)/q; Fa #FP
Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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![### Exercise 1.31: Finding an Element of a Given Order in a Finite Field
#### Problem Statement:
Let \( p \) be a prime number, and let \( q \) be another prime number that divides \( p - 1 \).
##### Part (a):
- Let \( \alpha \in \mathbb{F}_p^* \), the multiplicative group of non-zero elements of the finite field \( \mathbb{F}_p \).
- Define \( b = \alpha^{(p-1)/q} \). Prove that either \( b = 1 \) or \( b \) has order \( q \).
**Hint**: Recall that the order of \( b \) is the smallest integer \( k \geq 1 \) such that \( b^k = 1 \) in \( \mathbb{F}_p^* \). Use Proposition 1.30 for guidance.
##### Part (b):
- Suppose that we want to find an element of \( \mathbb{F}_p^* \) with order \( q \). Using part (a), we can randomly choose a value of \( \alpha \in \mathbb{F}_p^* \) and check whether \( b = \alpha^{(p-1)/q} \) satisfies \( b \neq 1 \).
- How likely are we to succeed? In other words, compute the value of the ratio:
\[
\frac{\{ \alpha \in \mathbb{F}_p^* : \alpha^{(p-1)/q} \neq 1 \}}{|\mathbb{F}_p^*|}
\]
**Hint**: Use Theorem 1.31 to aid in your calculation.
#### Explanation for Ratio Calculation:
To understand the likelihood of finding such an element \( b \neq 1 \) with the condition given, the goal is to evaluate the proportion of elements in \( \mathbb{F}_p^* \) that do not become 1 when raised to the power \((p-1)/q\). The formula provided represents this ratio, comparing the set of all such \( \alpha \in \mathbb{F}_p^* \) to the total number of elements in \( \mathbb{F}_p^* \).
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This modified text can be used as educational content to explain a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c5ad030-3ec8-4fd2-8d64-821b0d0d0877%2F341d8c9f-bcb7-4341-9c41-98957422e298%2Ft462oxa_processed.png&w=3840&q=75)
Transcribed Image Text:### Exercise 1.31: Finding an Element of a Given Order in a Finite Field
#### Problem Statement:
Let \( p \) be a prime number, and let \( q \) be another prime number that divides \( p - 1 \).
##### Part (a):
- Let \( \alpha \in \mathbb{F}_p^* \), the multiplicative group of non-zero elements of the finite field \( \mathbb{F}_p \).
- Define \( b = \alpha^{(p-1)/q} \). Prove that either \( b = 1 \) or \( b \) has order \( q \).
**Hint**: Recall that the order of \( b \) is the smallest integer \( k \geq 1 \) such that \( b^k = 1 \) in \( \mathbb{F}_p^* \). Use Proposition 1.30 for guidance.
##### Part (b):
- Suppose that we want to find an element of \( \mathbb{F}_p^* \) with order \( q \). Using part (a), we can randomly choose a value of \( \alpha \in \mathbb{F}_p^* \) and check whether \( b = \alpha^{(p-1)/q} \) satisfies \( b \neq 1 \).
- How likely are we to succeed? In other words, compute the value of the ratio:
\[
\frac{\{ \alpha \in \mathbb{F}_p^* : \alpha^{(p-1)/q} \neq 1 \}}{|\mathbb{F}_p^*|}
\]
**Hint**: Use Theorem 1.31 to aid in your calculation.
#### Explanation for Ratio Calculation:
To understand the likelihood of finding such an element \( b \neq 1 \) with the condition given, the goal is to evaluate the proportion of elements in \( \mathbb{F}_p^* \) that do not become 1 when raised to the power \((p-1)/q\). The formula provided represents this ratio, comparing the set of all such \( \alpha \in \mathbb{F}_p^* \) to the total number of elements in \( \mathbb{F}_p^* \).
---
This modified text can be used as educational content to explain a
![### Theorem 1.31 (Primitive Root Theorem)
Let \( p \) be a prime number. Then there exists an element \( g \in \mathbb{F}_p^* \) whose powers give every element of \( \mathbb{F}_p^* \), i.e.,
\[ \mathbb{F}_p^* = \{1, g, g^2, g^3, \ldots, g^{p-2}\}. \]
Elements with this property are called **primitive roots** of \( \mathbb{F}_p \) or generators of \( \mathbb{F}_p^* \). They are the elements of \( \mathbb{F}_p^* \) having order \( p - 1 \).
#### Proof
See [126, Chapter 20] or one of the texts [33, 47, 53, 90, 101].
The theorem is highlighting an important concept in finite field theory, particularly the existence of generators within the multiplicative group of a finite field. These generators are crucial for applications such as cryptography and the study of cyclic groups in abstract algebra.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c5ad030-3ec8-4fd2-8d64-821b0d0d0877%2F341d8c9f-bcb7-4341-9c41-98957422e298%2Fc8bq3xk_processed.png&w=3840&q=75)
Transcribed Image Text:### Theorem 1.31 (Primitive Root Theorem)
Let \( p \) be a prime number. Then there exists an element \( g \in \mathbb{F}_p^* \) whose powers give every element of \( \mathbb{F}_p^* \), i.e.,
\[ \mathbb{F}_p^* = \{1, g, g^2, g^3, \ldots, g^{p-2}\}. \]
Elements with this property are called **primitive roots** of \( \mathbb{F}_p \) or generators of \( \mathbb{F}_p^* \). They are the elements of \( \mathbb{F}_p^* \) having order \( p - 1 \).
#### Proof
See [126, Chapter 20] or one of the texts [33, 47, 53, 90, 101].
The theorem is highlighting an important concept in finite field theory, particularly the existence of generators within the multiplicative group of a finite field. These generators are crucial for applications such as cryptography and the study of cyclic groups in abstract algebra.
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