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Let $K=\mathbb{Z}(\sqrt{-5})$ be a field and $\mathcal{O}_K=\mathbb{Z}(\sqrt{-5})$ be the ring of integers. What are the ideals $\mathfrak{a} \subseteq \mathcal{O}_K=\mathbb{Z}(\sqrt{-5})$ with norm $N(\mathfrak{a})<100$? For a principal ideal $\mathfrak{a}=(a_1+a_2\sqrt{-5})$ the norm is $a_1^2+5a_2^2<100$ this region is bounded by an ellipse and so these ideals are straightforward to list. However, I know this field has class number two so this list must be incomplete.

Even a list of prime ideals might be more straightforward to write. We could list primes with $[p\mathbb{Z}(\sqrt{-5}):\mathbb{Z}(\sqrt{-5})]=p^2=100$ and some of these lattices will factor. However there are certainly other prime ideals $\mathfrak{p}\subseteq \mathbb{Z}(\sqrt{-5})$ that do not lie on the real number line $\mathbb{R}$.

On Wikipedia there's a fairly long list of Gaussian integers and their prime factorization in $\mathbb{Z}(i)$. However no such data set exists in this ring.

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  • $\begingroup$ Yes, you can start by listing prime ideals. It would be a good idea to remember their norms and then you just mix and match them to get the rest (the ones with composite norm). One could find the list of prime ideals using Kummer-Dedekind theorem, which (roughly) associates the factorization of x^2 + 5 mod p to prime ideals above p. $\endgroup$
    – JGA
    Jul 3, 2018 at 17:44
  • $\begingroup$ perhaps i need to get good at factoring math.stackexchange.com/q/320343 $\endgroup$
    – cactus314
    Jul 3, 2018 at 17:56

2 Answers 2

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You can consider how primes split in $\mathcal{O}_K=\mathbb{Z}(\sqrt{-5})$. If we try to find the discriminant since $-5 \equiv 3\pmod 4$, $D= 4 \cdot -5=-20$, and primes not dividing $-20$ split if $\left(\frac{-20}{p}\right)=\left(\frac{-4}{p}\right)\left(\frac{p}{5}\right)=1$. From there you should be able to get a set of congruence conditions for primes that split. As far as primes that don't split, you only need consider those primes $<10$, since otherwise $N(p)=p^2>100$. Finally $2$ and $5$ both ramify so primes of $\mathcal{O}_K$ lying above them also need to be considered but should be a short, finite check from there.

I get $\{2, 3, 5, 7, 23, 29, 37, 41, 43, 47, 61, 67, 83, 89\}$ as the list of primes with norm less than $100$ that ramify or split using the congruences $p \equiv 1, 3, 7, 9 \pmod {20}$ for primes that split in $\mathcal{O}_K$. There are no inert primes with norm $<100$.

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  • $\begingroup$ there should be other primes that lie off the real number line. that lie in $\mathbb{Z}[\sqrt{-5}]$ but not in $\mathbb{Z}$. $\endgroup$
    – cactus314
    Jul 3, 2018 at 18:25
  • $\begingroup$ @cactus314 those primes lie above the rational primes that split, so, for example, there are 2 primes lying above the rational prime 3, each with norm 3, or 2 primes lying above the rational prime 89. $\endgroup$
    – sharding4
    Jul 3, 2018 at 18:27
  • $\begingroup$ @cactus314 if you want all the ideals explicitly you would have to compute as suggested by JGA in his comment above. There is an ideal, call it $2_1$ lying above $2$ generated by $(2,1+\sqrt{-5})$, and there are primes $3_1=(3,1+\sqrt{-5})$ and $3_2=(3,2+\sqrt{-5})$ lying above $3$ none of which can be principal, otherwise they would be represented by the quadratic from $x^2+5y^2$. Likewise above 7 and 23. Finally with $29$ you get two primes $29_1=(29,13+\sqrt{-5})=(3-2\sqrt{-5})$ and $29_2=(29,16+\sqrt{-5})=(3+2\sqrt{-5})$ which are principal. And so on. $\endgroup$
    – sharding4
    Jul 3, 2018 at 19:19
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sage has an implementation for this. After defining K as below, one can ask details on the implementation via ?K.ideals_of_bdd_norm.

See also ideals_of_bdd_norm.

sage: K.<a> = QuadraticField(-5)
sage: K
Number Field in a with defining polynomial x^2 + 5
sage: for norm, ideals in K.ideals_of_bdd_norm(100).items():
....:     for J in ideals:
....:         print norm, J
....:         
1 Fractional ideal (1)
2 Fractional ideal (2, a + 1)
3 Fractional ideal (3, a + 2)
3 Fractional ideal (3, a + 1)
4 Fractional ideal (2)
5 Fractional ideal (-a)
6 Fractional ideal (-a + 1)
6 Fractional ideal (a + 1)
7 Fractional ideal (7, a + 4)
7 Fractional ideal (7, a + 3)
8 Fractional ideal (4, 2*a + 2)
9 Fractional ideal (-a - 2)
9 Fractional ideal (3)
9 Fractional ideal (a - 2)
10 Fractional ideal (10, a + 5)
12 Fractional ideal (6, 2*a + 4)
12 Fractional ideal (6, 2*a + 2)
14 Fractional ideal (a - 3)
14 Fractional ideal (a + 3)
15 Fractional ideal (15, a + 5)
15 Fractional ideal (15, a + 10)
16 Fractional ideal (4)
18 Fractional ideal (18, a + 11)
18 Fractional ideal (6, 3*a + 3)
18 Fractional ideal (18, a + 7)
20 Fractional ideal (-2*a)
21 Fractional ideal (2*a + 1)
21 Fractional ideal (a + 4)
21 Fractional ideal (a - 4)
21 Fractional ideal (-2*a + 1)
23 Fractional ideal (23, a + 15)
23 Fractional ideal (23, a + 8)
24 Fractional ideal (-2*a + 2)
24 Fractional ideal (2*a + 2)
25 Fractional ideal (5)
27 Fractional ideal (27, a + 20)
27 Fractional ideal (9, 3*a + 6)
27 Fractional ideal (9, 3*a + 3)
27 Fractional ideal (27, a + 7)
28 Fractional ideal (14, 2*a + 8)
28 Fractional ideal (14, 2*a + 6)
29 Fractional ideal (2*a + 3)
29 Fractional ideal (-2*a + 3)
30 Fractional ideal (-a - 5)
30 Fractional ideal (-a + 5)
32 Fractional ideal (8, 4*a + 4)
35 Fractional ideal (35, a + 25)
35 Fractional ideal (35, a + 10)
36 Fractional ideal (-2*a - 4)
36 Fractional ideal (6)
36 Fractional ideal (2*a - 4)
40 Fractional ideal (20, 2*a + 10)
41 Fractional ideal (a - 6)
41 Fractional ideal (a + 6)
42 Fractional ideal (42, a + 11)
42 Fractional ideal (42, a + 25)
42 Fractional ideal (42, a + 17)
42 Fractional ideal (42, a + 31)
43 Fractional ideal (43, a + 34)
43 Fractional ideal (43, a + 9)
45 Fractional ideal (2*a - 5)
45 Fractional ideal (-3*a)
45 Fractional ideal (2*a + 5)
46 Fractional ideal (-3*a + 1)
46 Fractional ideal (3*a + 1)
47 Fractional ideal (47, a + 29)
47 Fractional ideal (47, a + 18)
48 Fractional ideal (12, 4*a + 8)
48 Fractional ideal (12, 4*a + 4)
49 Fractional ideal (3*a - 2)
49 Fractional ideal (7)
49 Fractional ideal (-3*a - 2)
50 Fractional ideal (10, 5*a + 5)
54 Fractional ideal (a - 7)
54 Fractional ideal (-3*a + 3)
54 Fractional ideal (3*a + 3)
54 Fractional ideal (-a - 7)
56 Fractional ideal (2*a - 6)
56 Fractional ideal (2*a + 6)
58 Fractional ideal (58, a + 45)
58 Fractional ideal (58, a + 13)
60 Fractional ideal (30, 2*a + 10)
60 Fractional ideal (30, 2*a + 20)
61 Fractional ideal (3*a + 4)
61 Fractional ideal (-3*a + 4)
63 Fractional ideal (63, a + 11)
63 Fractional ideal (21, 3*a + 12)
63 Fractional ideal (63, a + 25)
63 Fractional ideal (63, a + 38)
63 Fractional ideal (21, 3*a + 9)
63 Fractional ideal (63, a + 52)
64 Fractional ideal (8)
67 Fractional ideal (67, a + 53)
67 Fractional ideal (67, a + 14)
69 Fractional ideal (2*a + 7)
69 Fractional ideal (a - 8)
69 Fractional ideal (a + 8)
69 Fractional ideal (-2*a + 7)
70 Fractional ideal (-3*a - 5)
70 Fractional ideal (3*a - 5)
72 Fractional ideal (36, 2*a + 22)
72 Fractional ideal (12, 6*a + 6)
72 Fractional ideal (36, 2*a + 14)
75 Fractional ideal (15, 5*a + 10)
75 Fractional ideal (15, 5*a + 5)
80 Fractional ideal (-4*a)
81 Fractional ideal (4*a - 1)
81 Fractional ideal (-3*a - 6)
81 Fractional ideal (9)
81 Fractional ideal (3*a - 6)
81 Fractional ideal (-4*a - 1)
82 Fractional ideal (82, a + 35)
82 Fractional ideal (82, a + 47)
83 Fractional ideal (83, a + 59)
83 Fractional ideal (83, a + 24)
84 Fractional ideal (4*a + 2)
84 Fractional ideal (2*a + 8)
84 Fractional ideal (2*a - 8)
84 Fractional ideal (-4*a + 2)
86 Fractional ideal (a - 9)
86 Fractional ideal (a + 9)
87 Fractional ideal (87, a + 74)
87 Fractional ideal (87, a + 16)
87 Fractional ideal (87, a + 71)
87 Fractional ideal (87, a + 13)
89 Fractional ideal (4*a - 3)
89 Fractional ideal (-4*a - 3)
90 Fractional ideal (90, a + 65)
90 Fractional ideal (30, 3*a + 15)
90 Fractional ideal (90, a + 25)
92 Fractional ideal (46, 2*a + 30)
92 Fractional ideal (46, 2*a + 16)
94 Fractional ideal (-3*a + 7)
94 Fractional ideal (3*a + 7)
96 Fractional ideal (-4*a + 4)
96 Fractional ideal (4*a + 4)
98 Fractional ideal (98, a + 81)
98 Fractional ideal (14, 7*a + 7)
98 Fractional ideal (98, a + 17)
100 Fractional ideal (10)
sage:
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  • $\begingroup$ software is fine... does "$a$" here measn $\sqrt{-5}$ ? Then we have $(2, 1+\sqrt{-5})$ and $(3, 1+\sqrt{-5}) \subseteq \mathbb{Z}[\sqrt{-5}]$ are prime ideals... It says fractional ideal but i'm guessing these are integers. $\endgroup$
    – cactus314
    Jul 4, 2018 at 13:30

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