# Rational Numbers form Subfield of Complex Numbers

## 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 $\struct {\C, + \times}$.

$\blacksquare$