×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite