# Proof

This proof is a direct application of the theorem construction of fields from integral domains $$( * )$$, in which we will construct the field $$(\mathbb Q, +,\cdot)$$ from the integral domain $$(\mathbb Z, +,\cdot)$$.

In the proof of the theorem $$( * )$$, a field $$(F,\ast,\circ)$$ is constructed from an integral domain $$(R, +,\cdot )$$, as the set of all equivalence classes $$[a,b]$$, which are represented by ordered pairs $$(a,b),(c,d)\in R\times (R\setminus\{0\})$$ together with the following equivalence relation:

$$(a,b)\sim (c,d)\Leftrightarrow ad=bc.$$

In this proof, the operations $$\ast$$ and $$\circ$$ are defined as follows: $$\begin{array}{ccl} \lbrack a,b\rbrack \ast\lbrack c,d\rbrack &:=&\lbrack ad + cb,~bd\rbrack ,\\ \lbrack a,b\rbrack \circ\lbrack c,d\rbrack &:=&\lbrack ac,~bd\rbrack .\\ \end{array}$$

For the special case of the integral domain $$(R, + ,\cdot) =(\mathbb Z, + ,\cdot )$$, the equivalence relation is defined for two ratios

$$\frac ab\sim \frac cd\Leftrightarrow ad=bc,~~~~~~~~~~a,c\in\mathbb Z,~~b,d\in\mathbb Z\setminus\{0\}$$

and the field operations in $$(\mathbb Q,\ast,\circ)$$, "addition $$\ast$$" and the "multiplication $$\circ$$" can be now written as $$\begin{array}{ccccc} \frac ab \ast \frac cd &:=&\frac ab + \frac cd &=&\frac {a\cdot d + c\cdot b}{b\cdot d},\\ \frac ab \circ\frac cd &:=&\frac ab \cdot \frac cd &=&\frac {a\cdot c}{b\cdot d}.\\ \end{array}$$ Thereby, we have replaced the notation $$(\mathbb Q,\ast,\circ)$$ by the more common notation $$(\mathbb Q, + ,\cdot)$$.

Furthermore, according to the proof of $$( * )$$, the integral domain $$(\mathbb Z, +,\cdot )$$ is isomorphic to a proper subset $$S\subset\mathbb Q$$ with (using our new notation) $$\mathbb Z\simeq S:=\left\{\frac {a\cdot x}x~|~a\in\mathbb Z,~x\in\mathbb Z\setminus\{0\}\right\}.$$ Please note, that since $$(\mathbb Z, +,\cdot )$$ is an integral domain, we can even write $$\frac {a\cdot x}x=\frac {a\cdot \cancel x}{\cancel x}=\frac a1$$. Thus, $$(\mathbb Z, +, \cdot)$$ can be regarded as a proper subset of $$(\mathbb Q, +,\cdot)$$ (i.e. each integer is a special case of a fraction), preserving the effect of the operations $$+$$ and $$\cdot$$, defined for fractions.

We also learn from the proof of theorem $$( * )$$:

• the additive identity (zero) is given by the fraction $$0:=\frac 01$$,
• the multiplicative identity is given by the fraction $$1:=\frac 11$$.
• the additive inverse element to a given fraction $$\frac ab$$ is $$-\frac ab:=\frac {-a}b$$, $$b\neq 0$$.
• the multiplicative inverse element to a given fraction $$\frac ab$$ is $$\frac ba$$, $$a,b\neq 0$$.

This completes the proof.

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### References

#### Bibliography

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