Intereting Posts

How do you prove this integral involving the Glaisher–Kinkelin constant
Odds of anyone in a group getting picked twice in a row
Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$
Continued fraction expansion related to exponential generating function
Path connectedness is a topological invariant?
Non-associative commutative binary operation
Why is the category of groups not closed, or enriched over itself?
What are the three non-isomorphic 2-dimensional algebras over $\mathbb{R}$?
Help solving a limit
why is the following thing a projection operator?
Is the Euler characteristic $\chi =2$ for the prism with a hole?
Good book for high school algebra
How to integrate this improper integral.
Tensor products of infinite-dimensional spaces and other objects
Equilateral triangle in a circle

Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4, p$ is a prime.

I know that $\pi$ is prime in $\mathbb Z[i]$ implies it is irreducible. Also if $\pi$ has a prime norm, that is $N(\pi) = p$ then $\pi$ is irreducible.

For any $z \in \mathbb Z[i]$ we have $N(z) = a^2 + b^2$. Since $\pi \notin \mathbb Z, i \mathbb Z$ both $a$ and $b$ must be non-zero in the case of $\pi$. By Fermat’s Two Square theorem every prime number $\equiv 1 \pmod 4$ can be written as a unique sum of two squares. I have some idea I should utilize this theorem here, but no success so far.

- Easy way to show that $\mathbb{Z}{2}]$ is the ring of integers of $\mathbb{Q}{2}]$
- Extending Herstein's Challenging Exercise to Modules
- Find the center of the symmetry group $S_n$.
- Converse to Chinese Remainder Theorem
- Why is this polynomial irreducible?
- Why “characteristic zero” and not “infinite characteristic”?

However I’ve come to a dead end. I don’t know how to go on from here. Could someone help me out ?

Thanks.

- A ring that is not a Euclidean domain
- Does the Frattini subgroup $\Phi(G)$ contain the intersection $Z(G)\cap $.
- Can you construct a field with 6 elements?
- Logic and number theory books
- How to count the number of solutions for this expression modulo a prime number $p$?
- Group actions in towers of Galois extensions
- What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation)
- Smooth functions for which $f(x)$ is rational if and only if $x$ is rational
- Can we rediscover the category of finite (abelian) groups from its morphisms?
- Intuitive definitions of the Orbit and the Stabilizer

By Fermat’s Two Square theorem every prime number $\equiv 1 \pmod{4}$ can be written as a unique sum of two squares.

So you know that rational positive primes $\equiv 1 \pmod{4}$ (and also $2$) are reducible in $\mathbb{Z}[i]$. You also know, or can easily check, that $\mathbb{Z}[i]$ is a Euclidean ring, hence a PID, hence a UFD.

Now consider $N(\pi) = \nu \in \mathbb{Z} \subset \mathbb{Z}[i]$. Let

$$\nu = \prod_{k=1}^r p_k^{\alpha_k}$$

be the prime factorisation in $\mathbb{Z}$. Let $p$ be a prime factor of $\nu$, and let $\pi_p$ be a prime element in $\mathbb{Z}[i]$ that divides $p$. Then

$$\pi_p \mid \nu = N(\pi) = \pi\overline{\pi} \Rightarrow (\pi_p \mid \pi) \lor (\overline{\pi_p} \mid \pi).$$

But that means $\pi \sim \pi_p$ or $\pi \sim \overline{\pi_p}$, in particular, $\pi \mid p$, whence $N(\pi) \mid N(p) = p^2$.

So there are two possibilities,

- $N(\pi) = p$, and that is what we want to show (you just need to say why $p \equiv 3\pmod{4}$ is impossible under the hypothesis).
- $N(\pi) = p^2$, but that would mean $\pi \sim p$, and $p$ itself would be prime in $\mathbb{Z}[i]$. You just need to say why that is impossible under the hypothesis.

Well…there you did all, didn’t you? I mean, *you* say you know $\;N(\pi)=a^2+b^2\;$ is a prime in $\;\Bbb Z\;$, and thus it either is $\;2\;$ or else an odd prime that can be expressed as the sum of two squares $\;\iff p\neq 3\pmod 4\iff p=1\pmod 4\;$ ….and voila!

Now, it isn’t true that $\;\alpha\in\Bbb Z[i]\;$ is a prime $\;\implies N(\alpha)\in\Bbb Z\;$ is a prime. For example, $\;N(7)=49\;$ , yet $\;7\in\Bbb Z[i]\;$ is a prime.

- Continued fraction of a square root
- Which is bigger among (i) $\log_2 3$ and $\log _3 5$ (ii) $\log_2 3$ and $\log _3 11$.
- Trapezoid rule error analysis
- Total number of solutions of an equation
- Formal power series ring, norm.
- What is the expected value of the number of die rolls necessary to get a specific number?
- What is the difference between “family” and “set”?
- How to show that $f$ is a straight line if $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$?
- (Unsolved) In this infinite sequence, no term is a prime: prove/disprove.
- A real function on a compact set is continuous if and only if its graph is compact
- Pollard-Strassen Algorithm
- How to prove these two ways give the same numbers?
- How is this proof correct in regard to a $k$-subalgebra (Eisenbud)?
- Fubini theorem for sequences
- Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$