\[
\DeclareMathOperator{\ker}{ker}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\char}{char}
\DeclareMathOperator{\lcm}{lcm}
\newcommand\divides{\mathbin |}
\newcommand\ndivides{\mathbin \nmid}
\newcommand\d{\mathrm{d}}
\newcommand\p{\partial}
\newcommand\C{\mathbb{C}}
\newcommand\N{\mathbb{N}}
\newcommand\Q{\mathbb{Q}}
\newcommand\R{\mathbb{R}}
\newcommand\Z{\mathbb{Z}}
\newcommand\pref[1]{(\ref{#1})}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Gal}{Gal}
\]

Return to table of contents for Brian's unofficial solutions
to Artin's *Algebra*
## Section 2.3. Subgroups of the Additive Group of Integers

Exercise 2.3.3(a) Let \(S\) be a nonempty,
possibly infinite set of integers. If \(S = \{0\}\), then define the GCD to be
0. Otherwise, the subgroup of \(\mathbb{Z}^+\) generated by \(S\) contains a
least positive element \(d\) and by Theorem 2.3.3, this subgroup is precisely
\(d\mathbb{Z}\). We define the GCD of \(S\) to be \(d\).