If p = 1 mod 4, then we can still define F,[i] as a ring, but it is not a field. Illustrate this in a specific case by finding a non-zero element in F[i] that is not invertible. Hint: in any ring, a zero divisor is an element x such that there is another non-zero element y with xy = 0. A zero divisor is never invertible – indeed, if x is is invertible then x xy -1 = x0 y = 0. xY = 0 Thus it suffices to find a non-zero zero divisor in F5|i].
If p = 1 mod 4, then we can still define F,[i] as a ring, but it is not a field. Illustrate this in a specific case by finding a non-zero element in F[i] that is not invertible. Hint: in any ring, a zero divisor is an element x such that there is another non-zero element y with xy = 0. A zero divisor is never invertible – indeed, if x is is invertible then x xy -1 = x0 y = 0. xY = 0 Thus it suffices to find a non-zero zero divisor in F5|i].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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I require help with this problem from Number theory
![If p = 1 mod 4, then we can still define Fp[i] as a ring, but it is not a field. Illustrate this in
a specific case by finding a non-zero element in F;[i] that is not invertible.
Hint: in any ring, a zero divisor is an element x such that there is another non-zero element
y with xy = 0. A zero divisor is never invertible
indeed, if x is is invertible then
x-lxy = x='0 = y = 0.
xy = 0
Thus it suffices to find a non-zero zero divisor in F5[i].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79ab4db3-2e61-48d8-a012-1335c209cbe0%2F7e5aa6df-da48-43a9-a91a-4f715b86a6dd%2Fd5ki7v3_processed.png&w=3840&q=75)
Transcribed Image Text:If p = 1 mod 4, then we can still define Fp[i] as a ring, but it is not a field. Illustrate this in
a specific case by finding a non-zero element in F;[i] that is not invertible.
Hint: in any ring, a zero divisor is an element x such that there is another non-zero element
y with xy = 0. A zero divisor is never invertible
indeed, if x is is invertible then
x-lxy = x='0 = y = 0.
xy = 0
Thus it suffices to find a non-zero zero divisor in F5[i].
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