P r o b l e m a : {\displaystyle Problema:} Fie K {\displaystyle K} un corp cu m ≥ 2 {\displaystyle m\geq 2} elemente si f ∈ K [ X ] {\displaystyle f\in K[X]} . Aratati ca urmatoarele afirmatii sunt echivalente:
(i) Exista g ∈ K [ X ] {\displaystyle g\in K[X]} astfel incat f ( X ) = g ( X m − 1 ) {\displaystyle f(X)=g(X^{m-1})} ;
(ii) Pentru orice a ∈ K ∗ {\displaystyle a\in K^{*}} avem f ( X ) = f ( a X ) {\displaystyle f(X)=f(aX)} .
S o l u t i e {\displaystyle Solutie} ( i ) → ( i i ) {\displaystyle (i)\rightarrow (ii)} Din teorema lui Lagrange aplicata grupului ( K ∗ , ⋅ ) {\displaystyle (K^{*},\cdot )} avem ca
![{\displaystyle f\in K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ec2d50f6317a5098e90e7e9742f89a1f133824)
![{\displaystyle g\in K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1414a4dff7c6064b41a4f7921cec55ef1c78a934)
