Question

prove that 5 is not prime element in the ring R of Gaussian integers​

Answer 1

Answer:

Tip:

• The Gaussian integers are the set of complex numbers whose real and imaginary parts are both integers.
• The Gaussian integers with ordinary addition and multiplication of complex numbers form an integral domain, usually written as$\mathbb{Z}[i]=\left \{ x+iy :x,y\in \mathbb{Z}\right \}$.

• $\mathbb{Z}[i]$ is a ring (really a subring of $\mathbb{C}$ ) since it is closed under addition and multiplication:$(x + iy) + (p + iq) = (x + p) + i(y + q), (x + iy)(p + iq) = (xp+yq) + i(xq + yp)$

To prove: $5$ is a prime element in the ring R of Gaussian integers​.

Explanation: Image result for proving that $5$ is not a prime element in the ring R of Gaussian integers​

We know that p is congruent to 1 modulo 4, then it is the product of a Gaussian prime by its conjugate, both of which are non-associated Gaussian primes.

i.e., neither is the product of the other by a unit.

Hence, p is said to be a decomposed prime in the Gaussian integers.

For example,

$5=\left (2+i)(2-i),3=(3+i)(3-i)$

Disclaimer: the above proof is for 5 is a prime element in the ring R of Gaussian integers​