Rational Numbers form Subfield of Complex Numbers

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Theorem

Let $\left({\Q, +, \times}\right)$ be the field of rational numbers.

Let $\left({\C, +, \times}\right)$ be the field of complex numbers.


Then $\left({\Q, +, \times}\right)$ is a subfield of $\left({\C, +, \times}\right)$.


Proof

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

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

Thus from Subfields Transitive $\left({\Q, +, \times}\right)$ is a subfield of $\left({\C, + \times}\right)$.

$\blacksquare$


Sources