Order of $\mathbb F_q^n$?

Question

Let $\mathbb F_q^n=\{ (x_1,x_2,\ldots,x_n)|x_1,\ldots,x_n \in \mathbb F_q\}$, with $q$ prime.

I know that:

$|\mathbb F_q^n|=q \times q \times \cdots \times q=q^n$

but what is a good, simple proof that this is the order? Sorry I feel like this should be simple but I'm a little stuck!

Answer 1

As a $\Bbb{F}_q$ vector space, $\Bbb{F}_q^n$ has dimension $n,$ so as an abelian group $$\Bbb{F}_q^n \cong \underbrace{\Bbb{F}_q\oplus \Bbb{F}_q\oplus \cdots \oplus \Bbb{F}_q}_{\text{$n$-times}}$$ and so $$|\Bbb{F}_q^n| =\underbrace{|\Bbb{F}_q| \cdots |\Bbb{F}_q|}_{\text{$n$-times}}=|\Bbb{F}_q|^n$$

$$=\underbrace{q\times q\times \cdots\times q}_{\text{$n$-times}}=q^n.$$