Proof
(related to Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication)
The set of rational numbers \(\mathbb Q\), together with the specific addition operation
"\(+\)", and the specific multiplication operation
"\(\cdot\)", forms the field the algebraic structure \((\mathbb Q, + , \cdot)\). This is because:
 We have shown that the set \((\mathbb Q, + )\) is a commutative group.
 We have shown that the set \((\mathbb Q^*, \cdot )\) is a commutative group, in which \(\mathbb Q^*\) denotes all nonzero rational numbers.
 We have shown that the set \((\mathbb Q^*, \cdot )\) is a commutative group, in which \(\mathbb Q^*\) denotes all nonzero rational numbers.
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References
Bibliography
 Kramer Jürg, von Pippich, AnnaMaria: "Von den natürlichen Zahlen zu den Quaternionen", SpringerSpektrum, 2013