Rings and Polynomials
A ring is an abelian additive group R that is equipped with an additional multiplication, just like the ring \(\mathbb Z\) of integers. More specifically, it is required that R be a monoid with respect to the multiplication and that the multiplication be distributive over the addition. We will always assume that the multiplication of a ring is commutative, except for a few occasions in Sect. 2.1. If the nonzero elements of a ring form an (abelian) group under the multiplication, the ring is actually a field. In principle, the definition of a ring goes back to R. Dedekind . For Dedekind, rings were motivated by questions in number theory involving integral elements in algebraic number fields, or in other words, by the study of algebraic equations with integer coefficients. However, we will deal with rings of integral algebraic numbers only occasionally. More important for us are fields serving as coefficient domains for algebraic equations, as well as polynomial rings over fields. These are of fundamental importance in studying algebraic equations, and in particular algebraic field extensions.
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