Abstract Algebra: Theory and Applications · Problem 33

Polynomials — Thomas Judson

Mathematics — Abstract Algebra · localization

Problem

Let pp be prime. Prove that Z(p)={a/b:a,b∈Z and gcd⁡(b,p)=1}\mathbb{Z}_{(p)} = \{a/b : a, b \in \mathbb{Z} \text{ and } \gcd(b, p) = 1\} is a ring. The ring Z(p)\mathbb{Z}_{(p)} is called the ring of integers localized at pp. Hint. Let a/b,c/d∈Z(p)a/b, c/d \in \mathbb{Z}_{(p)}. Then a/b+c/d=(ad+bc)/bda/b + c/d = (ad + bc)/bd and (a/b)⋅(c/d)=(ac)/(bd)(a/b) \cdot (c/d) = (ac)/(bd) are both in Z(p)\mathbb{Z}_{(p)}, since gcd⁡(bd,p)=1\gcd(bd, p) = 1.

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