Norms and primes

**JCIR** · April 29th 2008, 07:30 PM

I need an example of an irreducible number in form of a+bsqrt(p) that its norm is not prime.

**Isomorphism** · April 29th 2008, 10:27 PM

Originally Posted by JCIR

I need an example of an irreducible number in form of a+bsqrt(p) that its norm is not prime.

How do you define the norm of $a+b\sqrt{p}$ ?

Is it $\sqrt{a^2 + b^2}$ ?

**JCIR** · April 30th 2008, 03:40 AM

The norm is defined by N(a+bsqrt(p))= /a^2-pb^2/

**Moo** · April 30th 2008, 03:48 AM

Hello,

Is p prime ?

**JCIR** · April 30th 2008, 05:12 AM

p is prime or -1

**ThePerfectHacker** · April 30th 2008, 07:10 AM

Originally Posted by JCIR

I need an example of an irreducible number in form of a+bsqrt(p) that its norm is not prime.

Consider $\mathbb{Z}[i]$ and then any prime $p\equiv 3(\bmod 4)$ is a Gaussian prime and its norm is $p^2$ - not a prime.

More specifially, $3\in \mathbb{Z}[i]$ and $N(3) = 9$ with $3$ being irreducible.

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