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A cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to $\mathbb Q$, the field of rational numbers.

A cyclotomic field is the splitting field of the cyclotomic polynomial

$$ \Phi_n(x) = \prod_\stackrel{1\le k\le n}{\gcd(k,n)=1} \left(x-e^{2i\pi\frac{k}{n}}\right) $$

and therefore it is a Galois extension of the field of rational numbers. The degree of the extension

$$ [Q(ζ_n):Q]$$

is given by $φ(n)$ where $φ$ is Euler's phi function.

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