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[parent] cyclotomic field (Definition)

A cyclotomic field (or cyclotomic number field) is a cyclotomic extension of $\mathbb{Q}$ . These are all of the form $\mathbb{Q}(\omega_n)$ , where $\omega_n$ is a primitive $n$ th root of unity.

The ring of integers of a cyclotomic field always has a power basis over $\mathbb{Z}$ . Specifically, the ring of integers of $\mathbb{Q}(\omega_n)$ is $\mathbb{Z}[\omega_n]$ .

Given a primitive $n$ th root of unity $\omega_n$ , its minimal polynomial over $\mathbb{Q}$ is the cyclotomic polynomial $\Phi_n(x)$ . Thus, $[\mathbb{Q}(\omega_n)\!:\!\mathbb{Q}]=\varphi(n)$ , where $\varphi$ denotes the Euler phi function.

If $n$ is odd, then $\mathbb{Q}(\omega_{2n})=\mathbb{Q}(\omega_n)$ . There are many ways to prove this, but the following is a relatively short proof: Since $\omega_n={\omega_{2n}}^2\in \mathbb{Q}(\omega_{2n})$ , we have that $\mathbb{Q}(\omega_n)\subseteq\mathbb{Q}(\omega_{2n})$ . We also have that $[\mathbb{Q}(\omega_{2n})\!:\!\mathbb{Q}]=\varphi(2n)=\varphi(2)\varphi(n)=\varphi(n)=[\mathbb{Q}(\omega_n)\!:\!\mathbb{Q}]$ . Thus, $[\mathbb{Q}(\omega_{2n})\!:\!\mathbb{Q}(\omega_n)]=1$ . It follows that $\mathbb{Q}(\omega_{2n})=\mathbb{Q}(\omega_n)$ .

Note. If $n$ is a positive integer and $m$ is an integer such that $\gcd(m,n)=1$ , then $\omega_n$ and ${\omega_n}^m$ are primitive $n$ th roots of unity and generate the same cyclotomic field.




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See Also: cyclotomic extension, cyclotomic polynomial

Other names:  cyclotomic number field

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Cross-references: integer, positive, proof, odd, Euler phi function, cyclotomic polynomial, minimal polynomial, ring of integers, cyclotomic extension
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This is version 6 of cyclotomic field, born on 2007-05-30, modified 2007-06-24.
Object id is 9486, canonical name is CyclotomicField.
Accessed 2770 times total.

Classification:
AMS MSC11-00 (Number theory :: General reference works )
 11R18 (Number theory :: Algebraic number theory: global fields :: Cyclotomic extensions)

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