Show that $F_p ^2$ , considered as a vector space over Fp , is a union of finitely many proper subspaces.

Question

Show that $F_p ^2$ , considered as a vector space over Fp , is a union of finitely many proper subspaces. please explain what do u mean by $F_p ^2$ before answering the qstn. I know about $F_p$ as $Z$/p$Z$ and there are finite fields of p^n elements Also answer the qstn.

Answer 1

Ordered pairs $(a,b)$ of elements $a,b\in F_p$, equipeed with pointwise addition and scalar multiplication, i.e

$$c(a,b)=(ca,cb)$$ $$(a,b)+(c,d)=(a+c,b+d)$$

for every $a,b,c,d\in F_p$.

See ordered pairs of numbers. The construction is entirely similar, replacing the real numbers by $F_p$.

Answer 2

@JackozeeHakkiuz has provided the definition of that vector space. Here's a hint for proving that it is a union of finitely many subspaces: can you see why the real plane (real two dimensional vector space) is a union of lines through the origin?