Articles of gaussian integers

Proving Gaussian Integers are countable

I know that Gaussian Integers are a subset of complex numbers. They are numbers in the form G = {a + ib | a,b ∈ Z} So to prove that a set is countable, I need to find a function from G to N (Natural Numbers) s.t every n has finitely many preimages (from […]

Quotient field of gaussian integers

Let $D$ be the set of all gaussian integers of the form $m+ni$ where $m,n \in Z$. Carry out the construction of the quotient field $Q$ for this integral domain. Show that this quotient field is isomorphic to the set of all complex numbers of the form $a + bi$ where $a,b$ are rational numbers. […]

Algebraically, why is $\mathbb{Z}/(i + 1)$ isomorphic to $\mathbb{Z}_{2}$?

This question already has an answer here: Help with proof that $\mathbb Z[i]/\langle 1 – i \rangle$ is a field. 3 answers

Quotient ring of Gaussian integers $\mathbb{Z}/(a+bi)$ when $a$ and $b$ are NOT coprime

The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is well-known, when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d>1$? — For instance if $p$ is prime (which is not coprime with $0$) then $$\mathbb{Z}[i]/(p) \cong \mathbb{F}_p[X]/(X^2+1) \cong \begin{cases} \mathbb{F}_{p^2} &\text{if } p \equiv 3 \pmod 4\\ \mathbb{F}_{p} […]

Quotient ring of Gaussian integers

A very basic ring theory question, which I am not able to solve. How does one show that $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$. Extending the result: $\mathbb{Z}[i]/(a-ib) \cong \mathbb{Z}/(a^{2}+b^{2})\mathbb{Z}$, if $a,b$ are relatively prime. My attempt was to define a map, $\varphi:\mathbb{Z}[i] \to \mathbb{Z}/10\mathbb{Z}$ and show that the kernel is the ideal generated by $\langle{3-i\rangle}$. But I […]

Prove that the Gaussian Integer's ring is a Euclidean domain

I’m having some trouble proving that the Gaussian Integer’s ring ($\mathbb{Z}[ i ]$) is an Euclidean domain. Here is what i’ve got so far. To be a Euclidean domain means that there is a defined application (often called norm) that verifies this two conditions: $\forall a, b \in \mathbb{Z}[i] \backslash {0} \hspace{2 mm} a \mid […]

Any odd number is of form $a+b$ where $a^2+b^2$ is prime

This conjecture is tested for all odd natural numbers less than $10^8$: If $n>1$ is an odd natural number, then there are natural numbers $a,b$ such that $n=a+b$ and $a^2+b^2\in\mathbb P$. $\mathbb P$ is the set of prime numbers. I wish help with counterexamples, heuristics or a proof. Addendum: For odd $n$, $159<n<50,000$, there are […]