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 ) {\displaystyle (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})} ;
( i i ) {\displaystyle (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 x m − 1 = 1 , ∀ x ∈ K ∗ {\displaystyle x^{m-1}=1,\forall x\in K^{*}} , deci f ( a X ) = g ( ( a X ) m − 1 ) = g ( a m − 1 X m − 1 ) = g ( X m − 1 ) = f ( X ) {\displaystyle f(aX)=g((aX)^{m-1})=g(a^{m-1}X^{m-1})=g(X^{m-1})=f(X)} .
( i i ) → ( i ) {\displaystyle (ii)\rightarrow (i)}
![{\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)
