Factorization of Ideals in Number Field, Looking for Reference

jinydu · 2012-12-31, 06:22

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:

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.

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:

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.

jinydu · 2013-01-02, 18:09

Quote:

Originally Posted by Nick

Yes (but with $\mathbb{Z}$ instead of $\mathbb{Q}$ ):

Oops. Yes, silly me, thanks.

R.D. Silverman · 2014-07-30, 11:24

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.

2012-12-31, 06:22	#1
jinydu Dec 2003 Hopefully Near M48 1758₁₀ 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

2013-01-02, 06:17	#3
jinydu Dec 2003 Hopefully Near M48 2·3·293 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

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