Let Z[i] = {a + bi | a, b ≤ Z} be the ring of Gaussian integers.Determine how many cosets the principal ideal I = (2 — i) of Z[i] has and write down thetables for addition and multiplication of these cosets.

As per the question we have to find all the cosets that the principal ideal I = ⟨2 - i⟩ of ℤ[i] has…

Answered: Let Z[i] = {a + bi | a, b € Z} be the…

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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please, just answer the last 3. Thank you
If R= Z[x], f(x) = x2 +1, and g(x) = a0+a1x+...+anxn is a polynomial of degree 2 or more in R. How do you prove that g(x) is congruent to g(x)-anxn-anxn-2 mod I? I is the principal ideal generated by f(x)
Consider the proof below of the fact that I + a = 1 + b iff a - bEI where R is a commutative right with unity and I is an ideal. Identify which “+" signs are notation and which "+" signs are operations. (=) Assume I + a = 1 + b. : a E l + a : a el + b . a = i+ b for some i EI : i = a – b i a – b E l (=) Assume a – b E I Assume ΧΕ α :x = i + a for some i E I :x = i+ a + (-b+ b) :x = (i + a - b) + b i x E I + b : I + a CI+ b Assume x E I+ b :x = i + b for some i El :x = i +b+ (-a + b) : x = (i + b – a) + a : x € l + a :I+b SI+a : I+ a = 1 + b Consider the proof above and justify each claim.
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Let Z[i] = {a + bi | a, b € Z} be the ring of Gaussian integers. = Determine how many cosets the principal ideal I tables for addition and multiplication of these cosets. (2) of Z[i] has and write down the