Es gilt offensichtlich K ⊊ K [ x ] {\displaystyle K\subsetneq K[x]} und K [ x ] ⊆ L {\displaystyle K[x]\subseteq L} . Weiter gilt:
p = [ L : K ] = [ L : K [ x ] ] ⋅ [ K [ x ] : K ] {\displaystyle p=[L:K]=[L:K[x]]\cdot [K[x]:K]}
Da p {\displaystyle p} Primzahl ist, gilt entweder [ L : K [ x ] ] = 1 {\displaystyle [L:K[x]]=1} oder [ K [ x ] : K ] = 1 {\displaystyle [K[x]:K]=1} . Letzteres kann wegen K ⊊ K [ x ] {\displaystyle K\subsetneq K[x]} nicht gelten, also gilt [ L : K [ x ] ] = 1 {\displaystyle [L:K[x]]=1} und somit L = K [ x ] {\displaystyle L=K[x]} .
![{\displaystyle K\subsetneq K[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53604c5c6d5feff5214015dda330b27e4ac5fb0a)
![{\displaystyle K[x]\subseteq L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7719efadf456611ce8bdee2bff0e630aa9d29c2c)
![{\displaystyle p=[L:K]=[L:K[x]]\cdot [K[x]:K]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b90d728c372f4fa168bcad1ae8d3cdea42df33d)
![{\displaystyle [L:K[x]]=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7b0da952a2bd5b4d4ac6b2a8416a3ff9bd97b32)
![{\displaystyle [K[x]:K]=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b35847bc10efcbe0018c92a1cf1169c080f9927)
![{\displaystyle L=K[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a9dd47e7836efefd76dba284c034ff86a0abf4)