Real Numbers form Subfield of Complex Numbers

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Theorem

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


Proof

From Additive Group of Reals is Subgroup of Complex, $\struct {\R, +}$ is a subgroup of $\struct {\C, +}$.

From Multiplicative Group of Reals is Subgroup of Complex, $\struct {\R, \times}$ is a subgroup of $\struct {\C, \times}$.

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

$\blacksquare$


Sources