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 
[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. 
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=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. 
[QUOTE=Nick;323374]Yes (but with [tex]\mathbb{Z}[/tex] instead of [tex]\mathbb{Q}[/tex]):[/QUOTE]
Oops. Yes, silly me, thanks. 
[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. 
All times are UTC. The time now is 20:32. 
Powered by vBulletin® Version 3.8.11
Copyright ©2000  2021, Jelsoft Enterprises Ltd.