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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:

1. We have shown that the set \((\mathbb Q, + )\) is a commutative group.
2. We have shown that the set \((\mathbb Q^*, \cdot )\) is a commutative group, in which \(\mathbb Q^*\) denotes all non-zero rational numbers.
3. We have shown that the set \((\mathbb Q^*, \cdot )\) is a commutative group, in which \(\mathbb Q^*\) denotes all non-zero rational numbers.
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References

Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013