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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 [tex]\mathbb{Q}(2^{1/3})[/tex]. Find the factorizations of (7), (29) and (31) in [tex]O_K[/tex].

I know there's a theorem by Kronecker that says (7) is reducible iff [tex]x^3\equiv 2 \text{mod }7[/tex], 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 [tex]\mathbb{Q}[/tex]' and 'prime in [tex]\mathbb{Q}[/tex]' respectively? As opposed to 'element of [tex]\mathbb{O_K}[/tex]' and 'prime in [tex]\mathbb{O_K}[/tex]'?

Nick 2013-01-02 09:44

[QUOTE=jinydu;323361]Thanks. I presume 'rational integer' and 'rational prime' mean 'element of [tex]\mathbb{Q}[/tex]' and 'prime in [tex]\mathbb{Q}[/tex]' respectively? As opposed to 'element of [tex]\mathbb{O_K}[/tex]' and 'prime in [tex]\mathbb{O_K}[/tex]'?[/QUOTE]
Yes (but with [tex]\mathbb{Z}[/tex] instead of [tex]\mathbb{Q}[/tex]):
in algebraic number theory, the elements of [tex]\mathbb{Z}[/tex] 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 [tex]\mathbb{Z}[/tex] instead of [tex]\mathbb{Q}[/tex]):[/QUOTE]

Oops. Yes, silly me, thanks.

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

[QUOTE=jinydu;323160]Let K be the number field [tex]\mathbb{Q}(2^{1/3})[/tex]. Find the factorizations of (7), (29) and (31) in [tex]O_K[/tex].

I know there's a theorem by Kronecker that says (7) is reducible iff [tex]x^3\equiv 2 \text{mod }7[/tex], 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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