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Abstract Algebra

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Primes, Gaussian Integers, and Gaussian Primes

ABSALG-KTH7I4

A Gaussian integer is by definition a complex number $\alpha=x+iy$, where $i=\sqrt{-1}$ and $x,y\in{\mathbb Z}$, and the set of Gaussian integers is denoted ${\mathbb Z}[i]$.

We call the four Gaussian integers $1, -1, i, -i$ Gaussian units.

The four numbers $\alpha, -\alpha, i\alpha, -i\alpha$ are called the associates of $\alpha$, (not to be confused with the word ``associate'' used for the inverse in $({\mathbb Z}/p{\mathbb Z})^\ast$, $p$ prime).

We say that $\pi$ is prime in ${\mathbb Z}[i]$ if and only if, for every factorization $\pi=\alpha\beta$, $\alpha$, $\beta$ $\in {\mathbb Z}[i]$, we must have $\alpha$, $\beta$ either associates of $\pi$ or Gaussian units.

Which of the following is always true​?

A

A prime number $p$ in ${\mathbb Z}$ remains prime in ${\mathbb Z}[i]$.

B

A prime number $p$ in ${\mathbb Z}$ of the form $4m+1$ remains prime in ${\mathbb Z}[i]$.

C

A prime number $p$ in ${\mathbb Z}$ of the form $4m+3$ remains prime in ${\mathbb Z}[i]$.

D

If $\pi$ is prime in ${\mathbb Z}[i]$, then its complex conjugate $\overline{\pi}$ is not prime in ${\mathbb Z}[i]$.

E

For every prime $\pi$ in ${\mathbb Z}[i]$ the product $\pi{\overline{\pi}}$ of $\pi$ with its complex conjugate $\overline{\pi}$ is prime in ${\mathbb Z}$.