(b) Note that for a E R, v(a) E N. Show that R is an Euclidean domain with respect to the norm v. (c) Show that a is a unit in R if and only if v(a) = 1.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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(b) Note that for a ER, v(a) E N. Show that R is an Euclidean domain with respect
to the norm v.
(c) Show that a is a unit in R if and only if v(a) = 1.
Transcribed Image Text:(b) Note that for a ER, v(a) E N. Show that R is an Euclidean domain with respect to the norm v. (c) Show that a is a unit in R if and only if v(a) = 1.
Let R = Z[√2] = {a+b√2: a, b € Z} and F = Q[√2] = {a+b√2: a,b ≤ Q}.
Then R is a ring, F is a field, and RC FCR.
Define the norm on F by setting v(a+b√2) = |a² − 2b21. Note that for a = a +b√2,
v(a) = |aa| where a = a - b√2
Transcribed Image Text:Let R = Z[√2] = {a+b√2: a, b € Z} and F = Q[√2] = {a+b√2: a,b ≤ Q}. Then R is a ring, F is a field, and RC FCR. Define the norm on F by setting v(a+b√2) = |a² − 2b21. Note that for a = a +b√2, v(a) = |aa| where a = a - b√2
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