Polynomials and primitive elements in Galois rings
For a prime \(p\) and integer \(n\geq 1\), let \(\mathbb{Z}_{p^ n}\) denote the ring of integers modulo \(p^ n\), and let \(g(x)\in\mathbb{Z}[x]\) be a monic integral polynomial which is irreducible when viewed as a polynomial over the finite field \(\mathbb{Z}_ p\). The Galois ring \(GR(p^ n,m)\) is a ring of order \(p^{mn}\) which can be viewed as the set of all expressions \(A(\alpha)=a_ 0+a_ 1\alpha+a_ 2\alpha^ 2+\cdots+a_{m- 1}\alpha^{m-1}\), \(a_ i\in\mathbb{Z}_{p^ n}\), where the addition of such expressions is componentwise modulo \(p^ n\) and the multiplication is modulo \(g(x)\). Note that \(GR(p^ n,1)\) is the ring \(\mathbb{Z}_{p^ n}\) and \(GR(p^ 1,m)\) is the finite field \(GF(p^ m)\). Also note that if \(\beta\) is a root of \(g(x)\) in \(GF(p^ m)\), and if \(a\to\overline a\) is the natural homomorphism from \(\mathbb{Z}_{p^ n}\) to \(\mathbb{Z}_ p\), then \[ A(\alpha)\to\overline{A(\alpha)}=\overline a_ 0+\overline a_ 1\beta+\cdots+\overline a_{m-1}\beta^{m-1} \] is a homomorphism of \(GR(p^ n,m)\) to \(GF(p^ m)\). This mapping induces a natural mapping \({f(x)\to\overline f(x)}\) from \(GR(p^ n,m)[x]\) to \(GF(p^ m)[x]\). The main results of this paper concern the primitive elements of \(GR(p^ n,m)\) which, by definition, are those elements \(\gamma\) in the group of units \(U\) of \(GR(p^ n,m)\) with order equal to the exponent of \(U\). The main results are the following two theorems which are generalizations to Galois rings of known theorems for finite fields: Theorem. Let \(c\) be a positive integer and let \(Q\) denote the set of all odd prime powers \(q=p^ m\) for which \(c\) divides \(p^ m-1\). Then for all but finitely many elements of \(Q\) the following statement holds: if \(f(x)\) is a polynomial over \(GR(p^ n,m)\) such that \(\overline f(x)\) is square-free and has no zeros in \(GF(p^ m)\), then there exists a primitive element \(\gamma\) in \(GR(p^ n,m)\) such that \(f(\gamma)\) is a \(c\)-th power in \(GR(p^ n,m)\). Theorem. For all but finitely many Galois rings \(GR(p^ n,m)\), the following holds: if \(f(x)\) is a nonconstant polynomial over \(GR(p^ n,m)\) so that (i) \(\overline f(x)\) is square-free and (ii) the derivative of \(\overline f(x)\) does not vanish on \(GF(p^ m)\), then there exists a primitive element \(\alpha\) in \(GR(p^ n,m)\) such that \(f(\alpha)\) is a primitive element in \(GR(p^ n,m)\).
scientific article; zbMATH DE number 89649