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2.3. Let g be a primitive root for Fp. (a) Suppose that x = a and x = b are both integer solutions to the congruence g* = h (mod p). Prove that a = b (mod p - 1). Explain why this implies that the map (2.1) on page 63 is well-defined. (b) Prove that (c) Prove that log, (h₁h₂) = logg (h₁) + logg (h₂) log, (h): = n n logg (h) for all h₁, h2 € Fp. for all h E F and n € Z.

2.3. Let g be a primitive root for Fp. (a) Suppose that x = a and x = b are both integer solutions to the congruence g* = h (mod p). Prove that a = b (mod p - 1). Explain why this implies that the map (2.1) on page 63 is well-defined. (b) Prove that (c) Prove that log, (h₁h₂) = logg (h₁) + logg (h₂) log, (h): = n n logg (h) for all h₁, h2 € Fp. for all h E F and n € Z.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### 2.3. Let \( g \) be a primitive root for \( \mathbb{F}_p \). (a) Suppose that \( x = a \) and \( x = b \) are both integer solutions to the congruence \( g^x \equiv h \pmod{p} \). Prove that \( a \equiv b \pmod{p-1} \). Explain why this implies that the map (2.1) on page 63 is well-defined. (b) Prove that \( \log_g(h_1 h_2) = \log_g(h_1) + \log_g(h_2) \) for all \( h_1, h_2 \in \mathbb{F}_p^* \). (c) Prove that \( \log_g(h^n) = n \log_g(h) \) for all \( h \in \mathbb{F}_p^* \) and \( n \in \mathbb{Z} \).
Transcribed Image Text:### 2.3. Let \( g \) be a primitive root for \( \mathbb{F}_p \). (a) Suppose that \( x = a \) and \( x = b \) are both integer solutions to the congruence \( g^x \equiv h \pmod{p} \). Prove that \( a \equiv b \pmod{p-1} \). Explain why this implies that the map (2.1) on page 63 is well-defined. (b) Prove that \( \log_g(h_1 h_2) = \log_g(h_1) + \log_g(h_2) \) for all \( h_1, h_2 \in \mathbb{F}_p^* \). (c) Prove that \( \log_g(h^n) = n \log_g(h) \) for all \( h \in \mathbb{F}_p^* \) and \( n \in \mathbb{Z} \).
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