(4) Determine ker o. 5) Prove that the quotient ring Z[√-1] In. is a field if and only if q = n² + 1 is prim

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Chapter2: Second-order Linear Odes
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(4) Determine ker þ.
(5) Prove that the quotient ring
Z[√-1]
In
is a field if and only if q = n² + 1 is prime.
Transcribed Image Text:(4) Determine ker þ. (5) Prove that the quotient ring Z[√-1] In is a field if and only if q = n² + 1 is prime.
The following assignment will use repeatedly the following ring
Z[√−1] = {r + √-1s: r, s € Z},
the Gaussian integers. This is a subset of C and it is a commutative ring. (You don't need to prove these properties
you can assume them.)
Throughout this question let 0 ‡n € Z and let In = z[√-1)(√-1 − n). Set q = n² + 1 and denote the equivalence
class of rE Z mod q by [r].
Transcribed Image Text:The following assignment will use repeatedly the following ring Z[√−1] = {r + √-1s: r, s € Z}, the Gaussian integers. This is a subset of C and it is a commutative ring. (You don't need to prove these properties you can assume them.) Throughout this question let 0 ‡n € Z and let In = z[√-1)(√-1 − n). Set q = n² + 1 and denote the equivalence class of rE Z mod q by [r].
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