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The polynomial f(x) = x³ − x−3_mod (47) has x₁ = 10 as a root. Use Hensel's Lemma to lift this root to x₂ a root modulo 47² and also to root to x3 a root modulo 47³. Since (ƒ'(x₁)) = 17_mod (47), which is not zero, the root will lift to a unique root each time according to the formula Xn+1 = Xn − ƒ (Xn) (ƒ' (Xn))¯¹ Write the lifted roots as the standard representative of its congruence class; a value from the set {0, 1,..., (p² − 1)} for the first answer and a value from the set {0, 1,..., (p³ − 1)} for the second answer. - The lifted root modulo ² is The lifted root modulo p³ is

The polynomial f(x) = x³ − x−3_mod (47) has x₁ = 10 as a root. Use Hensel's Lemma to lift this root to x₂ a root modulo 47² and also to root to x3 a root modulo 47³. Since (ƒ'(x₁)) = 17_mod (47), which is not zero, the root will lift to a unique root each time according to the formula Xn+1 = Xn − ƒ (Xn) (ƒ' (Xn))¯¹ Write the lifted roots as the standard representative of its congruence class; a value from the set {0, 1,..., (p² − 1)} for the first answer and a value from the set {0, 1,..., (p³ − 1)} for the second answer. - The lifted root modulo ² is The lifted root modulo p³ is

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 92E
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The polynomial f(x) = x³ - x − 3 mod (47) has x₁ = 10 as a root. Use Hensel's Lemma to lift this root to x₂ a root modulo 47² and also to root to X3 a root modulo 47³. Since (f'(x₁)) = 17 mod (47), which is not zero, the root will lift to a unique root each time according to the formula Xn+1 = Xn − ƒ (xn) (ƒ'(xn))¯¹. Write the lifted roots as the standard representative of its congruence class; a value from the set {0, 1,..., (p² − 1)} for the first answer and a value from the set {0, 1,…..‚ (p³ – 1)} for the second answer. The lifted root modulo ² is The lifted root modulo p³ is
Transcribed Image Text:The polynomial f(x) = x³ - x − 3 mod (47) has x₁ = 10 as a root. Use Hensel's Lemma to lift this root to x₂ a root modulo 47² and also to root to X3 a root modulo 47³. Since (f'(x₁)) = 17 mod (47), which is not zero, the root will lift to a unique root each time according to the formula Xn+1 = Xn − ƒ (xn) (ƒ'(xn))¯¹. Write the lifted roots as the standard representative of its congruence class; a value from the set {0, 1,..., (p² − 1)} for the first answer and a value from the set {0, 1,…..‚ (p³ – 1)} for the second answer. The lifted root modulo ² is The lifted root modulo p³ is
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