对于哪个d，𝐐(√d) 上的整数分解不唯一？

hbghlyj

$𝐐(\sqrt{10})$上:
\[6=2 \cdot 3=(4+\sqrt{10})(4-\sqrt{10}) \text {. }\]类似地，
\begin{array}{lll}
\mathbf{Q}(\sqrt{15})  : & 10=2 \cdot 5=(5+\sqrt{15})(5-\sqrt{15}) \\
\mathbf{Q}(\sqrt{26})  : & 10=2 \cdot 5=(6+\sqrt{26})(6-\sqrt{26}) \\
\mathbf{Q}(\sqrt{30})  : & 6=2 \cdot 3=(6+\sqrt{30})(6-\sqrt{30})
\end{array}(以上例子并不是唯一的)

这个$d$的序列与oeis.org/A029702相同吗？

hbghlyj

1-100所有没有平方因数的 d:
[1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97]

1. # Define a function to check if a number is square-free
2. def is_square_free(n):
3.     # Factorize the number into prime powers
4.     factors = factor(n)
5.     # Check if any of the prime powers has an exponent greater than 1
6.     for p, e in factors:
7.         if e > 1:
8.             return False
9.     # If none of the prime powers has an exponent greater than 1, the number is square-free
10.     return True
12. # Create an empty list to store the square-free integers
13. square_free = []
15. # Loop through the numbers from 1 to 100
16. for n in range(1, 101):
17.     # Check if the number is square-free and add it to the list if it is
18.     if is_square_free(n):
19.         square_free.append(n)
21. # Print the list of square-free integers
22. print(square_free)

Copy the Code

如何从这个列表中过滤掉满足唯一分解条件的d？

睡神

太高深了，我连题目都还没看懂…还是先解决我的那个问题吧，至少我的更通熟易懂

hbghlyj

睡神 发表于 2024-2-19 14:12
我连题目都还没看懂

例如$𝐐(\sqrt{-1})$
$x^2+1=(x+i)(x-i)$
$𝐙(\sqrt{-1})$:
$2=-i(1+i)^2$

hbghlyj

wolframalpha.com/input?i=PrimeQ%5B1%2BI%5D
Input

Result

$1 + i$ is a prime number

wolframalpha.com/input?i=PrimeQ%5B7%2B24I%5D
Input

Result

$7+24i$ is not a prime number
$|7+24i|=25=|15+20i|$

Czhang271828

答案应该是数列 oeis.org/A219361 的补. 因为非唯一分解 Dedekind 整环的类数可以是 $2$ 以及更高, 例如 $\mathbb Q(\sqrt{79})$ 的代数整数环.

关于 $\mathcal O_{\mathbb Q(\sqrt{d})}$ 是否是唯一分解环: $d<0$ 时参考 Heegner numbers, $d>0$ 时没有简单的解.

hbghlyj

Czhang271828 发表于 2024-2-20 09:26
数列 oeis.org/A219361 的补.

由此找到oeis.org/A146209正是它的补

hbghlyj

Czhang271828 发表于 2024-2-20 09:26
例如 $\mathbb Q(\sqrt{79})$ 的代数整数环.

NumberFieldClassNumber[Sqrt[79]]=3

Dedekind domain $\mathbb{Z}[\sqrt{79}]$ has class group of order 3, representatives $(1), (3,\sqrt{79}+1), (3, \sqrt{79}-1)$.

相关：If $\mathcal{O}_{\Bbb{Q}(\sqrt{d})}$ has class number $2$ or higher, does that mean $\sqrt{d}$ is irreducible but not prime?