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(a) For $ \ n \in \mathbb{N} \ $ , let $ \ \zeta_n=e^{\large \frac{2 \pi \large i}{n}} \in \mathbb{C} \ $, Prove that $ \ \mathbb{Q}(\zeta_n) /\mathbb{Q} \ $ is finite.

(b) Let $ \ \mathbb{Q}(\zeta_{\infty}) =\cup_{n \in \mathbb{N}} \mathbb{Q}(\zeta_n) \subset \mathbb{C} \ $. Prove that $ \ \mathbb{Q}(\zeta_{\infty}) \subset \mathbb{C} \ $ is a subfield of $ \ \mathbb{C} \ $.

Answer:

(a)

Here $ \ \zeta_n \ $ is the primitive $ \ n^{th} \ $ root of unity.

We know that $ \ [\mathbb{Q}(\zeta_n): \mathbb{Q}]=\phi(n) \ $ , weher $ \ \phi \ $ is euler phi function.

Since $ \ \phi(n) \ $ is finite , we can conclude that $ \ \mathbb{Q}(\zeta_n) /\mathbb{Q} \ $ is finite.

I have still confusion.

But I can not answer part $ \ (b) \ $

Help me with part (b) and also verify part (a) answer

MAS
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1 Answers1

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Your solution to (a) seems to say "$\Bbb Q(\zeta_n)/\Bbb Q$ is a finite extension since we already know it's a finite extension".

For (b), the crucial observation is that for $m$ and $n$ there is some $t$ such that $\Bbb Q(\zeta_m)\cup\Bbb Q(\zeta_n)\subseteq\Bbb Q(\zeta_t)$.

Can you find a possible $t$?

Angina Seng
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  • Is $ \ t=mn \ $ ? where $ \ m,n \ $ are coprime – MAS Aug 07 '18 at 11:58
  • Indeed, $mn$ works! @yourmath – Angina Seng Aug 07 '18 at 12:43
  • Then how $ \ \mathbb{Q}(\zeta_{\infty}) \ $ would be a subfield of $ \ \mathbb{C} \ $? I can 't conclude it. Help me – MAS Aug 07 '18 at 14:07
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    To show it's closed under addition/multiplication, take two elements: one is in $\Bbb Q(\zeta_m)$ the other in $\Bbb Q(\zeta_n)$ for some $m$, $n$. Their sum and product must be in $\Bbb Q(\zeta_{mn})$. – Angina Seng Aug 07 '18 at 15:03