I understand how to factorize ideals into prime ideals when they are of the form $(p)$, by Dedekind's Theorem, but I can't factorize ideals like $(4+\sqrt{10})$ in $\mathbb{Z}[\sqrt{10}]$. I can calculate its norm: $16-10=6$, so I know any prime factors must be of norm $2$ and $3$, but I can't figure out how to find them except by trial and error. Any help would be really appreciated. Thanks in advance.

  • 1
    $\begingroup$ Seems that some people do not like British English... (by editing to replace "factorise" with "factorize") $\endgroup$ – Jérémy Blanc Mar 29 '14 at 13:06

If $\frak p$ is an ideal of norm $p$ prime in $\Bbb Z[\sqrt{10}]$, one knows that ${\frak p}\subset(p)=p\Bbb Z[\sqrt{10}]$, i.e. $\frak p$ appears in the decomposition of the ideal $(p)$.

The preliminary listing of primes ramifying, splitting or remaining inert helps. For instance if you know that $p$ is inert, then there are no ideals of norm $p$. The behaviour of the prime $p$ is easily linked to the behaviour of the polynomial $X^2-10\bmod p$.

All of this remains unchanged after replacing $\Bbb Z[\sqrt{10}]$ with the ring of integers of any quadratic extension, and in fact works--in principle--also for other algebraic extensions.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.