Cyclotomic Fields....

**AAM** · May 17th 2009, 10:12 AM

Hi everyone,

How can I prove that Q(Zeta36) = Q(Zeta12,Zeta18) does not contain ANY primitive 5th roots of unity?

[here, ZetaN = cos2pi/N + isin2pi/N ]

Many many thanks in advance. x

**ThePerfectHacker** · May 17th 2009, 11:33 AM

Originally Posted by AAM

Hi everyone,

How can I prove that Q(Zeta36) = Q(Zeta12,Zeta18) does not contain ANY primitive 5th roots of unity?

[here, ZetaN = cos2pi/N + isin2pi/N ]

Many many thanks in advance. x

In general $\mathbb{Q}(\zeta_{\ell}) = \mathbb{Q}(\zeta_n,\zeta_m)$ where $\ell = \text{lcm}(n,m)$ .
To prove this show each is a subset of eachother.

**AAM** · May 17th 2009, 11:46 AM

Hi perfecthacker - I think you may have misread my question - I wasn't asking about the equality of those 2 extensions - I was asking how to show that such an extension contains no primitive roots of unity.

Many thanks. :-)

**ThePerfectHacker** · May 17th 2009, 12:06 PM

Originally Posted by AAM

Hi perfecthacker - I think you may have misread my question - I wasn't asking about the equality of those 2 extensions - I was asking how to show that such an extension contains no primitive roots of unity.

Many thanks. :-)

Okay I fixed it.

It can be shown that $\mathbb{Q}(\zeta_n)\cap \mathbb{Q}(\zeta_m) = \mathbb{Q}(\zeta_d)$ where $d=\gcd(n,m)$ . Do you know this result? I can prove it for you.

If $\zeta_5\in \mathbb{Q}(\zeta_{36})$ then $\mathbb{Q}(\zeta_5)\subseteq \mathbb{Q}(\zeta_{36})$ and so $\mathbb{Q}(\zeta_5) \cap \mathbb{Q}(\zeta_{36}) = \mathbb{Q}(\zeta_5)$ .
But this is a problem because $\gcd(36,5) = 1$ .

By the way you can generalize this to prove that $\mathbb{Q}(\zeta_n)\subseteq \mathbb{Q}(\zeta_m)$ if and only if $n|m$ .

**AAM** · May 17th 2009, 04:10 PM

Beautiful! :-D

Thank you SO much perfecthacker! :-)

**AAM** · May 17th 2009, 04:13 PM

One minor point though - you have done the question for Zeta_5 = cos2pi/5+isin2pi/5.

But the original question was for Zeta = ANY PRIMITIVE 5th root of unity.

So my question is does Q(Zeta) = Q(Zeta_5) ? Because if so then your argument still holds. :-)

**ThePerfectHacker** · May 17th 2009, 07:04 PM

Originally Posted by AAM

Beautiful! :-D

Thank you SO much perfecthacker! :-)

Yes, I am beautiful.
But I am all alone,

, no girl wants to spend time with me.

Originally Posted by AAM

One minor point though - you have done the question for Zeta_5 = cos2pi/5+isin2pi/5.

But the original question was for Zeta = ANY PRIMITIVE 5th root of unity.

So my question is does Q(Zeta) = Q(Zeta_5) ? Because if so then your argument still holds. :-)

Let $\omega$ be a 5-th root of unity then the other primitive roots of unity are: $\omega,\omega^2,\omega^3, \omega^4$ . Therefore, $\zeta_5 \in \{ \omega,\omega^2,\omega^3,\omega^4\}$ . This means if $\omega \in \mathbb{Q}_{36}$ then $\{\omega,\omega^2,\omega^3,\omega^4\} \subset \mathbb{Q}_{36}$ and so $\zeta_5 \in \mathbb{Q}_{36}$ . Now apply the argument above.

This is Mine 12,5

th Post!!!

Thread: Cyclotomic Fields....

LinkBack

Thread Tools

Display

Cyclotomic Fields....

Similar Math Help Forum Discussions

vector fields - graphing gradient fields

primitive pth root of unity and cyclotomic fields

Cyclotomic Fields....

Cyclotomic Fields over Q

Cyclotomic Fields & Bases....

Search Tags