Wholly Real Number and Wholly Imaginary Number are Linearly Independent over the Rationals

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Theorem

Let $z_1$ be a non-zero wholly real number.

Let $z_2$ be a non-zero wholly imaginary number.


Then, $z_1$ and $z_2$ are linearly independent over the rational numbers $\Q$, where the group is the complex numbers $\C$.


Proof

From Rational Numbers form Subfield of Complex Numbers, the unitary module $\left({\C, +, \times}\right)_\Q$ over $\Q$ satisfies the unitary module axioms:


Let $a, b \in \Q$ such that:

$a z_1 + b z_2 = 0$

By the definition of wholly imaginary, there is a real number $c$ such that:

$c i = z_2$

where $i$ is the imaginary unit.

Therefore:

$a z_1 + b c i = 0$

Equating real parts and imaginary parts:

$a z_1 = 0$
$b c = 0$

Since $z_1$ and $c$ are both non-zero, $a$ and $b$ must be zero.

The result follows from the definition of linear independence.

$\blacksquare$