Factorization of quadratic polynomials in the ring of formal power series over $\Z$

Details Factorization of quadratic polynomials in the ring of formal power series over $\Z$ Factorization of quadratic polynomials in the ring of formal power series over $\Z$

2007-06-05

arxiv.org/abs/0706.0725v1

Mathematics

Commutative Algebra

15 pages

Scientific paper

We establish necessary and sufficient conditions for a quadratic polynomial
to be irreducible in the ring $Z[[x]]$ of formal power series with integer
coefficients. For $n,m\ge 1$ and $p$ prime, we show that $p^n+p^m\beta x+\alpha
x^2$ is reducible in $Z[[x]]$ if and only if it is reducible in $Z_p[x]$, the
ring of polynomials over the $p$-adic integers.

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