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-   -   Factorization of Ideals in Number Field, Looking for Reference (https://www.mersenneforum.org/showthread.php?t=17621)

 jinydu 2012-12-31 06:22

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

 Nick 2012-12-31 19:49

[QUOTE=jinydu;323160]
Can anyone suggest a reference? No spoilers to this problem please, just a reference.
Thanks[/QUOTE]
You could try "Problems in Algebraic Number Theory" by Murty & Esmonde
(Springer GTM 190) theorem 5.5.1.

 jinydu 2013-01-02 06:17

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 $$\mathbb{O_K}$$'?

 Nick 2013-01-02 09:44

[QUOTE=jinydu;323361]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 $$\mathbb{O_K}$$'?[/QUOTE]
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.

 jinydu 2013-01-02 18:09

[QUOTE=Nick;323374]Yes (but with $$\mathbb{Z}$$ instead of $$\mathbb{Q}$$):[/QUOTE]

Oops. Yes, silly me, thanks.

 R.D. Silverman 2014-07-30 11:24

[QUOTE=jinydu;323160]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[/QUOTE]

Henri Cohen's book.

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