Real Numbers form Subfield of Complex Numbers

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Theorem

The field of real numbers $\left({\R, +, \times}\right)$ forms a subfield of the field of complex numbers $\left({\C, +, \times}\right)$.


Proof

From Additive Group of Reals is Subgroup of Complex, $\left({\R, +}\right)$ is a subgroup of $\left({\C, +}\right)$.

From Multiplicative Group of Reals is Subgroup of Complex, $\left({\R, \times}\right)$ is a subgroup of $\left({\C, \times}\right)$.

The result follows from the Subfield Test via the One-Step Subgroup Test and Two-Step Subgroup Test.

$\blacksquare$


Sources