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Abstract Algebra

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Factorization in Gaussian Integers: Primes = Sum of Two Squares

ABSALG-89NB9E

Let $\mathbb{Z}[\sqrt{-1}]$ be the ring of Gaussian integers $\{a+b\sqrt{-1}\mid a, b\in\mathbb{Z}\}$.

Let $p\ge 3$ be a prime number in $\mathbb{Z}$ and suppose that $p$ is a sum of two squares:

$p=a^{2}+b^{2}$ for some $a$, $b\in\mathbb{Z}$

Select "True" or "False" for the following statements about such a prime $p$.

True

False

True

False

$p$ is congruent to $3$ mod $4$.

True

False

$p$ is the product of exactly two non-units in the ring $\mathbb{Z}[\sqrt{-1}]$.

True

False

$p$ is also prime in the ring $\mathbb{Z}[\sqrt{-1}]$.

True

False

If $p$, $p'$ are primes that are both the sum of two squares, then their product $pp'$ is not prime, but is still the sum of two squares.