ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sat, 07 Dec 2013 10:12:06 +0100why (4*i+7) in Z[i] not a prime?norm(4*i+7) =65, 65mod4=1https://ask.sagemath.org/question/10804/why-4i7-in-zi-not-a-primenorm4i7-65-65mod41/ K.<a> = NumberField(x^2 +1);
(4*a+7).norm()
65
mod (65,4)
1
factor(4*a+7)
(-3*a - 2) * (a - 2)
(4*a+7).multiplicative_order();
+Infinity
K.elements_of_norm(65);factor(-8*a + 1);K.prime_factors(4*a -7);
-8*a - 1, 4*a - 7, 4*a + 7, -8*a + 1]
(a - 2) * (3*a - 2)
[Fractional ideal (3*a - 2), Fractional ideal (-a - 2)]
[Fractional ideal (a - 2), Fractional ideal (3*a - 2)]
Fri, 06 Dec 2013 01:28:51 +0100https://ask.sagemath.org/question/10804/why-4i7-in-zi-not-a-primenorm4i7-65-65mod41/Answer by kcrisman for <pre><code>K.<a> = NumberField(x^2 +1);
(4*a+7).norm()
65
mod (65,4)
1
factor(4*a+7)
(-3*a - 2) * (a - 2)
(4*a+7).multiplicative_order();
+Infinity
K.elements_of_norm(65);factor(-8*a + 1);K.prime_factors(4*a -7);
-8*a - 1, 4*a - 7, 4*a + 7, -8*a + 1]
(a - 2) * (3*a - 2)
[Fractional ideal (3*a - 2), Fractional ideal (-a - 2)]
[Fractional ideal (a - 2), Fractional ideal (3*a - 2)]
</code></pre>
https://ask.sagemath.org/question/10804/why-4i7-in-zi-not-a-primenorm4i7-65-65mod41/?answer=15766#post-id-15766This is because 65 is not prime. Note that $1^2+2^5$ and $3^2+2^2=13$ are both primes (of the form $4n+1$, so the factors you find are indeed (Gaussian) primes.
You may find the [Wikipedia article](http://en.wikipedia.org/wiki/Gaussian_integer#As_a_principal_ideal_domain) helpful.Fri, 06 Dec 2013 10:45:07 +0100https://ask.sagemath.org/question/10804/why-4i7-in-zi-not-a-primenorm4i7-65-65mod41/?answer=15766#post-id-15766Comment by cjsh for <p>This is because 65 is not prime. Note that $1^2+2^5$ and $3^2+2^2=13$ are both primes (of the form $4n+1$, so the factors you find are indeed (Gaussian) primes.</p>
<p>You may find the <a href="http://en.wikipedia.org/wiki/Gaussian_integer#As_a_principal_ideal_domain">Wikipedia article</a> helpful.</p>
https://ask.sagemath.org/question/10804/why-4i7-in-zi-not-a-primenorm4i7-65-65mod41/?comment=15769#post-id-15769thank you very nuch!Sat, 07 Dec 2013 02:23:25 +0100https://ask.sagemath.org/question/10804/why-4i7-in-zi-not-a-primenorm4i7-65-65mod41/?comment=15769#post-id-15769Comment by kcrisman for <p>This is because 65 is not prime. Note that $1^2+2^5$ and $3^2+2^2=13$ are both primes (of the form $4n+1$, so the factors you find are indeed (Gaussian) primes.</p>
<p>You may find the <a href="http://en.wikipedia.org/wiki/Gaussian_integer#As_a_principal_ideal_domain">Wikipedia article</a> helpful.</p>
https://ask.sagemath.org/question/10804/why-4i7-in-zi-not-a-primenorm4i7-65-65mod41/?comment=16554#post-id-16554Great! In order to let future users know that was the correct answer, please accept it (there should be a check mark of some kind below the thumbs up/down symbols).Sat, 07 Dec 2013 10:12:06 +0100https://ask.sagemath.org/question/10804/why-4i7-in-zi-not-a-primenorm4i7-65-65mod41/?comment=16554#post-id-16554