Es sei K n {\displaystyle {}K_{n}} der n {\displaystyle {}n} -te Kreisteilungskörper über Q {\displaystyle {}\mathbb {Q} } . Wegen ( e 2 π i / n ) n = 1 {\displaystyle {}{\left(e^{2\pi {\mathrm {i} }/n}\right)}^{n}=1} ist Q [ e 2 π i / n ] ⊆ K n {\displaystyle {}\mathbb {Q} [e^{2\pi {\mathrm {i} }/n}]\subseteq K_{n}} . Wegen ( e 2 π i / n ) k = e 2 π i k / n {\displaystyle {}{\left(e^{2\pi {\mathrm {i} }/n}\right)}^{k}=e^{2\pi {\mathrm {i} }k/n}} gehören auch alle anderen Einheitswurzeln zu Q [ e 2 π i / n ] {\displaystyle {}\mathbb {Q} [e^{2\pi {\mathrm {i} }/n}]} , also ist Q [ e 2 π i / n ] = K n {\displaystyle {}\mathbb {Q} [e^{2\pi {\mathrm {i} }/n}]=K_{n}} .
![{\displaystyle {}\mathbb {Q} [e^{2\pi {\mathrm {i} }/n}]\subseteq K_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/318575a9a9df381ec43af702522354bc1d139997)
![{\displaystyle {}\mathbb {Q} [e^{2\pi {\mathrm {i} }/n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/246c0bcb2ec55d485d332858b47ccd53452e4863)
![{\displaystyle {}\mathbb {Q} [e^{2\pi {\mathrm {i} }/n}]=K_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09b41856b277e5aa9a815ac6a036b3a1a2d1e4a9)