Olympiad problems

1999 Romania National Olympiad, 4

Let $A$ be an integral domain and $A[X]$ be its associated ring of polynomials. For every integer $n \ge 2$ we define the map $\varphi_n : A[X] \to A[X],$ $\varphi_n(f)=f^n$ and we assume that the set $$M= \Big\{ n \in \mathbb{Z}_{\ge 2} : \varphi_n \mathrm{~is~an~endomorphism~of~the~ring~} A[X] \Big\}$$ is nonempty. Prove that there exists a unique prime number $p$ such that $M=\{p,p^2,p^3, \ldots\}.$

2018 Ramnicean Hope, 3

[b]a)[/b] Let $ u $ be a polynom in $ \mathbb{Q}[X] . $ Prove that the function $ E_u:\mathbb{Q}[X]\longrightarrow\mathbb{Q}[X] $ defined as $ E_u(P)=P(u) $ is an endomorphism. [b]b)[/b] Let $ E $ be an injective endomorphism of $ \mathbb{Q} [X] . $ Show that there exists a nonconstant polynom $ v $ in $ \mathbb{Q}[X] $ such that $ E(P)=P(v) , $ for any $ P $ in $ \mathbb{Q}[X] . $ [b]c)[/b] Let $ A $ be an automorphism of $ \mathbb{Q}[X] . $ Demonstrate that there is a nonzero constant polynom $ w $ in $ \mathbb{Q}[X] $ which has the property that $ A(P)=P(w) , $ for any $ P $ in $ \mathbb{Q}[X] . $ [i]Marcel Țena[/i]

function automorphism endomorphism morphism ring theory polynomial ring algebra

2010 Laurențiu Panaitopol, Tulcea, 4

Let be a ring $ R $ which has the property that there exist two distinct natural numbers $ s,t $ such that for any element $ x $ of $ R, $ the equation $ x^s=x^t $ is true. Show that there exists a polynom in $ R[X] $ of degree $$ |s-t|\left( 1+|s-t| \right) $$ such that all the elements of $ R $ are roots of it.

abstract algebra polynomial ring theory polynomial ring algebra polynomials