Algebraic number field basis

**Banach** · November 23rd 2008, 03:25 AM

Hi! I tried to show the following but i failed: We have a finite galois extension $L/\mathbb{Q}$ with commutative galois group and $O_L$ the ring of algebraic integers in L. Suppose there exists $x \in O_L$ so that the set $\{\sigma(x): \sigma \in$ Gal( $L/\mathbb{Q})\}$ forms a $\mathbb{Z}$ -Basis of $O_L$ .
Then for every field $K \subset L$ there also exists $y \in O_K$ so that $\{\phi(y): \phi \in$ Gal( $K/\mathbb{Q})\}$ forms a $\mathbb{Z}$ -Basis of $O_K$ .

I could show that $K/\mathbb{Q}$ is normal and hence each $\phi \in$ Gal $(K/\mathbb{Q})$ is just a restriction of $\sigma \in$ Gal $(L/\mathbb{Q})$ . But i dont know how to continue the argument. Can anybody please help me?

Greetings
Banach

**ThePerfectHacker** · November 23rd 2008, 10:37 AM

Originally Posted by Banach

forms a $\mathbb{Z}$ -Basis of $O_L$ .

I want to help but I do not what a Z-basis is.
Can you tell me?

**Banach** · November 23rd 2008, 10:57 AM

Hi! It is very nice that you want to help me. Here it just means that every $a \in O_L$ can be written as $a=\sum \limits_{i=1}^n \lambda_i \sigma_i(x)$ with $\lambda_i \in \mathbb{Z}$ and $\sigma_i$ the elements of the galois group.

**Banach** · November 25th 2008, 10:03 PM

Another idea: It also follows that K is galois, so maybe the primitive element theorem helps?

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