P10.2. Let w := }(−1 + √√3i) € C be one of the complex roots of the polynomial x² + x + 1 E in C[x]. Prove that the commutative domain Z[w] := {a + bw | a, b € Z} C C is a Euclidean domain with the function v: Z[w] →N defined by v(a + bw) = a²-ab+b².
P10.2. Let w := }(−1 + √√3i) € C be one of the complex roots of the polynomial x² + x + 1 E in C[x]. Prove that the commutative domain Z[w] := {a + bw | a, b € Z} C C is a Euclidean domain with the function v: Z[w] →N defined by v(a + bw) = a²-ab+b².
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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![P10.2. Let w := (-1 + √√3i) € C be one of the complex roots of the polynomial x² + x + 1
in C[x]. Prove that the commutative domain Z[w] := {a + bw | a,b ≤ Z} c C is a
Euclidean domain with the function v: Z[w] →N defined by v(a+bw)=a²-ab+b².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96191c65-e923-461d-9658-eb756d234e0f%2Fb4e07003-4d20-4e41-9eb9-018a94c4aac6%2Fnmiizfd_processed.png&w=3840&q=75)
Transcribed Image Text:P10.2. Let w := (-1 + √√3i) € C be one of the complex roots of the polynomial x² + x + 1
in C[x]. Prove that the commutative domain Z[w] := {a + bw | a,b ≤ Z} c C is a
Euclidean domain with the function v: Z[w] →N defined by v(a+bw)=a²-ab+b².
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