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 $\struct {\C, +, \times}_\Q$ over $\Q$ satisfies the unitary module axioms:
- $\Q$-Action: $\C$ is closed under multiplication, so $\Q \times \C \subset \C$.
- Distributive: $\times$ distributes over $+$.
- Associativity: $\times$ is associative.
- Multiplicative Identity: $1$ is the multiplicative identity in $\C$.
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.
$\blacksquare$