Prime Elements and Irreducible Polynomials over Some Imaginary Quadratic Fields
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A classical result of A. Cohn states that, if we express a prime $p$ in base $10$ as $$p=a_n10^n+a_{n-1}10^{n-1}+\cdots +a_110+a_0,$$ then the polynomial $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0$is irreducible in $\mathbb Z[x]$. This problem was subsequently generalized to any base $b$ by Brillhart, Filaseta, and Odlyzko. We establish this result of A. Cohn in $O_K[x]$, $K$ an imaginary quadratic field such that its ring of integers, $O_K$, is a Euclidean domain. For a Gaussian integer $\beta$ with $|\beta|>1+\sqrt{2}/2,$ we give another representation for any Gaussian integer using a complete residue system modulo $\beta,$ and then establish an irreducibility criterion in $\mathbb Z[i][x]$ by applying this result.
Abstract
A classical result of A. Cohn states that, if we express a prime $p$ in base $10$ as $$p=a_n10^n+a_{n-1}10^{n-1}+\cdots +a_110+a_0,$$ then the polynomial $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0$is irreducible in $\mathbb Z[x]$. This problem was subsequently generalized to any base $b$ by Brillhart, Filaseta, and Odlyzko. We establish this result of A. Cohn in $O_K[x]$, $K$ an imaginary quadratic field such that its ring of integers, $O_K$, is a Euclidean domain. For a Gaussian integer $\beta$ with $|\beta|>1+\sqrt{2}/2,$ we give another representation for any Gaussian integer using a complete residue system modulo $\beta,$ and then establish an irreducibility criterion in $\mathbb Z[i][x]$ by applying this result.
참고문헌 (9)
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