polynomial ring over integral domain
Theorem.
If the coefficient ring is an integral domain, then so is also its polynomial ring .
Proof. Let and be two non-zero polynomials in and let and be their leading coefficients, respectively. Thus , , and because has no zero divisors, . But the product is the leading coefficient of and so cannot be the zero polynomial. Consequently, has no zero divisors, Q.E.D.
Remark. The theorem may by induction be generalized for the polynomial ring .
Title | polynomial ring over integral domain |
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Canonical name | PolynomialRingOverIntegralDomain |
Date of creation | 2013-03-22 15:10:06 |
Last modified on | 2013-03-22 15:10:06 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 10 |
Author | pahio (2872) |
Entry type | Theorem |
Classification | msc 13P05 |
Related topic | RingAdjunction |
Related topic | FormalPowerSeries |
Related topic | ZeroPolynomial2 |
Related topic | PolynomialRingOverFieldIsEuclideanDomain |
Defines | coefficient ring |