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 2012-12-31, 06:22 #1 jinydu     Dec 2003 Hopefully Near M48 2×3×293 Posts Factorization of Ideals in Number Field, Looking for Reference Let K be the number field $\mathbb{Q}(2^{1/3})$. Find the factorizations of (7), (29) and (31) in $O_K$. I know there's a theorem by Kronecker that says (7) is reducible iff $x^3\equiv 2 \text{mod }7$, has a solution (or something like that) and how to find the factorization in the case it does have a solution. But I can't seem to find a reference for this. Can anyone suggest a reference? No spoilers to this problem please, just a reference. Thanks
2012-12-31, 19:49   #2
Nick

Dec 2012
The Netherlands

3×587 Posts

Quote:
 Originally Posted by jinydu Can anyone suggest a reference? No spoilers to this problem please, just a reference. Thanks
You could try "Problems in Algebraic Number Theory" by Murty & Esmonde
(Springer GTM 190) theorem 5.5.1.

 2013-01-02, 06:17 #3 jinydu     Dec 2003 Hopefully Near M48 33368 Posts Thanks. I presume 'rational integer' and 'rational prime' mean 'element of $\mathbb{Q}$' and 'prime in $\mathbb{Q}$' respectively? As opposed to 'element of $\mathbb{O_K}$' and 'prime in $\mathbb{O_K}$'? Last fiddled with by jinydu on 2013-01-02 at 06:19
2013-01-02, 09:44   #4
Nick

Dec 2012
The Netherlands

3·587 Posts

Quote:
 Originally Posted by jinydu Thanks. I presume 'rational integer' and 'rational prime' mean 'element of $\mathbb{Q}$' and 'prime in $\mathbb{Q}$' respectively? As opposed to 'element of $\mathbb{O_K}$' and 'prime in $\mathbb{O_K}$'?
Yes (but with $\mathbb{Z}$ instead of $\mathbb{Q}$):
in algebraic number theory, the elements of $\mathbb{Z}$ are called rational integers to distinguish them from algebraic integers, and similarly with primes.

2013-01-02, 18:09   #5
jinydu

Dec 2003
Hopefully Near M48

2×3×293 Posts

Quote:
 Originally Posted by Nick Yes (but with $\mathbb{Z}$ instead of $\mathbb{Q}$):
Oops. Yes, silly me, thanks.

2014-07-30, 11:24   #6
R.D. Silverman

Nov 2003

22·5·373 Posts

Quote:
 Originally Posted by jinydu Let K be the number field $\mathbb{Q}(2^{1/3})$. Find the factorizations of (7), (29) and (31) in $O_K$. I know there's a theorem by Kronecker that says (7) is reducible iff $x^3\equiv 2 \text{mod }7$, has a solution (or something like that) and how to find the factorization in the case it does have a solution. But I can't seem to find a reference for this. Can anyone suggest a reference? No spoilers to this problem please, just a reference. Thanks
Henri Cohen's book.

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