Rational Numbers form Subfield of Complex Numbers

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Theorem

Let $\struct {\Q, +, \times}$ denote the field of rational numbers.

Let $\struct {\C, +, \times}$ denote the field of complex numbers.


$\struct {\Q, +, \times}$ is a subfield of $\struct {\C, +, \times}$.


Proof

From Rational Numbers form Subfield of Real Numbers, $\struct {\Q, +, \times}$ is a subfield of $\struct {\R, + \times}$.

From Real Numbers form Subfield of Complex Numbers, $\struct {\R, +, \times}$ is a subfield of $\struct {\C, + \times}$.


Thus from Subfield of Subfield is Subfield $\struct {\Q, +, \times}$ is a subfield of $\struct {\C, + \times}$.

$\blacksquare$


Sources