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 $$[\mathbb Q(ζ_n):\mathbb Q]$$ is given by $$φ(n)$$, where $$φ$$ is Euler's phi function.