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prime factorization in $\mathbb{Z}[i]$ [duplicate]
We were asked to show where the following reasoning goes wrong. Since $1+i$ and $1-i$ are prime elements in $\mathbb{Z}[i]$, the equation $$(-i)(1+i)^2=(1+i)(1-i)=2$$ show that unique prime ...
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Is there an effective way to decompose gaussian integers into prime factors?
We define $\mathbb{Z}[i] := \{a + bi \mid a, b \in \mathbb{Z}\}, i = \sqrt{-1},$ which is an euclidean ring together with $N: \mathbb{Z}[i] \to \mathbb{N}_0, z \mapsto z\bar{z}=a^2+b^2$ for $z=a+bi$. ...
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Let $\mathbb{Z}[i]$ denote the *Gaussian integers*. Factor both $3+i$ and its norm into primes in $\mathbb{Z}[i]$
Question: Let $\mathbb{Z}[i]$ denote the Gaussian integers.
(a) Compute the norm $N(3+i)$ of $3+i$ in $\mathbb{Z}[i]$
(b) Factor both $3+i$ and its norm into primes in $\mathbb{Z}[i]$
(c) Compute $\...
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A question about Gaussian integers [duplicate]
I am working on the following exercise:
Show that if $p \equiv 1 \mod 4$ then $p$ is not prime in $\mathbb{Z}[i]$, but instead splits as the product of two distinct prime. [Hint: Show that $p|(a^2+1)$ ...
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