Factorization of Quadratic Polynomials in the Ring of Formal Power Series Over
نویسنده
چکیده
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 ≥ 1 and p prime, we show that p+pβx+αx is reducible in Z[[x]] if and only if it is reducible in Zp[x], the ring of polynomials over the p-adic integers.
منابع مشابه
Factorization of Quadratic Polynomials in the Ring of Formal Power Series over Z
We establish necessary and sufficient conditions for a quadratic polynomial to be irre-ducible in the ring Z[[x]] of formal power series over the integers. In particular, for polynomials of the form p n + p m βx + αx 2 with n, m ≥ 1 and p prime, we show that reducibility in Z[[x]] is equivalent to reducibility in Zp[x], the ring of polynomials over the p-adic integers.
متن کاملHYPERTRANSCENDENTAL FORMAL POWER SERIES OVER FIELDS OF POSITIVE CHARACTERISTIC
Let $K$ be a field of characteristic$p>0$, $K[[x]]$, the ring of formal power series over $ K$,$K((x))$, the quotient field of $ K[[x]]$, and $ K(x)$ the fieldof rational functions over $K$. We shall give somecharacterizations of an algebraic function $fin K((x))$ over $K$.Let $L$ be a field of characteristic zero. The power series $finL[[x]]$ is called differentially algebraic, if it satisfies...
متن کاملArithmetic in the Ring of Formal Power Series with Integer Coefficients
The divisibility and factorization theory of the integers and of the ring of polynomials (in one variable) over the integers are standard topics in a first course in abstract algebra. Concepts such as prime, irreducible, and invertible elements, unique factorization, and irreducibility criteria are extensively studied and are part of the core of the course. On the other hand, the natural extens...
متن کاملOn Hensel’s Roots and a Factorization Formula in Z[[x]]
Given an odd prime p, we provide formulas for the Hensel lifts of polynomial roots modulo p, and give an explicit factorization over the ring of formal power series with integer coe cients for certain reducible polynomials whose constant term is of the form pw with w > 1. All of our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange.
متن کاملA factorization formula in Z[[x]]
Given an odd prime p, we give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose constant term is of the form p with w > 1. Our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange. Résumé. Donné un nombre premier impair p, nous donnons une factorisation explicite sur l...
متن کامل