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Old 2012-12-31, 06:22   #1
jinydu
 
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Default 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
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Old 2012-12-31, 19:49   #2
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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.
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Old 2013-01-02, 06:17   #3
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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
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Old 2013-01-02, 09:44   #4
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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.
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Old 2013-01-02, 18:09   #5
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Yes (but with \mathbb{Z} instead of \mathbb{Q}):
Oops. Yes, silly me, thanks.
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Old 2014-07-30, 11:24   #6
R.D. Silverman
 
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Quote:
Originally Posted by jinydu View Post
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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