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On linear polynomials commutable with a polynomial. (Über die mit einem Polynom vertauschbaren linearen Polynome.) (German) Zbl 0741.12003
Let $$K[x]$$ be the polynomial ring over a field $$K$$, and $$\langle K[x],\circ\rangle$$ the semigroup under composition. The authors investigate the so-called linear centralizer $$LZ(f(x))=L\cap Z(f(x))$$, $$L$$ denoting the group of units of $$K[x]$$, $$L=\{ax+b,a,b,\in K,a\neq 0\}$$ and $$Z(f(x))$$ being the centralizer of $$f(x)$$, $$Z(f(x))=\{g(x)\in K[x]\backslash K$$, $$f(x)\circ g(x)=g(x)\circ f(x)\}$$. It is shown that there exists an $$\ell(x)\in L$$ such that for $$f_ 1(x)=\ell^{- 1}(x)\circ f(x)\circ\ell(x)$$ $$LZ(f_ 1(x))=T\circ S$$, where $$T,S$$ are subgroups of $$\{x+a,a\in K\}$$ and $$\{bx,b\in K^*\}$$, respectively. Furthermore it is proved that mostly $$T$$ is trivial, in particular if $$\text{char }K=0$$. The case $$\text{char }K=p$$ is dealt with, too.
Reviewer: G.Kowol (Wien)

##### MSC:
 12E05 Polynomials in general fields (irreducibility, etc.) 12E10 Special polynomials in general fields 13B25 Polynomials over commutative rings 20E99 Structure and classification of infinite or finite groups