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Provide a justification for each step in the proof outlined below showing that: The commutative ring (Z₁, +, *) with identity is not a field if n is a composite number (i.e., n is not prime). Recall that Zn = {0, 1, 2, 3, ..., n-1} with + and * representing modular addition and multiplication, respectively. Proof: a. If n is a composite number, then n = a*b, with 0

Provide a justification for each step in the proof outlined below showing that: The commutative ring (Z₁, +, *) with identity is not a field if n is a composite number (i.e., n is not prime). Recall that Zn = {0, 1, 2, 3, ..., n-1} with + and * representing modular addition and multiplication, respectively. Proof: a. If n is a composite number, then n = a*b, with 0

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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Provide a justification for each step in the proof outlined below showing that: The commutative ring (Z₁, +, *) with identity is not a field if n is a composite number (i.e., n is not prime). Recall that Zn = {0, 1, 2, 3, ..., n-1} with + and * representing modular addition and multiplication, respectively. Proof: a. If n is a composite number, then n = a*b, with 0<a<n and 0<b<n b. So, a*b =0 So, (Zn, +, *) is not a field Provide a justification for each step in the proof outlined below showing that: The commutative ring (Zn, +, *) with identity is a field if n is a prime number. Recall that Zn = {0, 1, 2, 3, ..., n-1} with + and * representing modular addition and multiplication, respectively. Proof: For any non-zero element, 'a', in Zn, gcd(a,n) = 1 b. There exists integers x and y (in Zn) that solve the Diophantine equation ax + ny = 1 (refer to the Number Theory course) c. But, ny = 0, so ax = 1 d. So, the number x is a's multiplicative inverse So, (Zn, +, *) is a field
Transcribed Image Text:Provide a justification for each step in the proof outlined below showing that: The commutative ring (Z₁, +, *) with identity is not a field if n is a composite number (i.e., n is not prime). Recall that Zn = {0, 1, 2, 3, ..., n-1} with + and * representing modular addition and multiplication, respectively. Proof: a. If n is a composite number, then n = a*b, with 0<a<n and 0<b<n b. So, a*b =0 So, (Zn, +, *) is not a field Provide a justification for each step in the proof outlined below showing that: The commutative ring (Zn, +, *) with identity is a field if n is a prime number. Recall that Zn = {0, 1, 2, 3, ..., n-1} with + and * representing modular addition and multiplication, respectively. Proof: For any non-zero element, 'a', in Zn, gcd(a,n) = 1 b. There exists integers x and y (in Zn) that solve the Diophantine equation ax + ny = 1 (refer to the Number Theory course) c. But, ny = 0, so ax = 1 d. So, the number x is a's multiplicative inverse So, (Zn, +, *) is a field
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