Integral Multiple of Integral Multiple
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Theorem
Let $\struct {F, +, \times}$ be a field.
Let $a \in F$ and $m, n \in \Z$.
Then:
- $\paren {m n} \cdot a = m \cdot \paren {n \cdot a}$
where $n \cdot a$ is as defined in integral multiple.
Proof
We have that $\struct {F^*, \times}$ is the multiplicative group of $\struct {F, +, \times}$.
Let $a \in F^*$, that is, $a \in F: a \ne 0_F$, where $0_F$ is the zero of $F$.
This is an instance of Powers of Group Elements when expressed in additive notation:
- $\forall m, n \in \Z: \paren {m n} a = m \paren {n a}$
$\Box$
Now suppose $a = 0_F$.
Then by definition of the zero element of $F$, we have that:
- $\paren {m n} \cdot a = 0_F = m \cdot \paren {n \cdot a}$
$\blacksquare$
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties