Finite extension of finite field

Question

I have two questions:

1. Suppose $F$ is a finite field then prove that it is a simple extension of its prime subfield.
2. Suppose $F$ is a finite extension of a finite field $K$. Then prove that $F$ is a simple extension of $K$.

I write proof of (1) but not sure if it is true or not. Please consider it. For (2) I felt that (1) may be helpful but I am not getting the proof.

Proof of (1). $F$ is a finite field. Then the extension $F|(\mathbb{Z}/p\mathbb{Z})$ is a finite extension where $\mathbb{Z}/p\mathbb{Z}$ is the prime subfield of $F.$ We know that $F^*$ is cyclic. Let $F^*=\langle u\rangle=\{1,u,u^2,\cdots,u^{n-1}\}$ therefore, $F=\{0,1,\cdots,u^{n-1}\}.$ Since $F|(\mathbb{Z}/p\mathbb{Z})$ is a finite dimensional vector space, we must have $a_ku^k+\cdots+a_0=0,a_i\in \mathbb{Z}/p\mathbb{Z},1\leq i\leq k.$ Therefore, $u$ is algebraic over $\mathbb{Z}/p\mathbb{Z}$ and $F=\mathbb{Z}/p\mathbb{Z}(u).$

Answer 1

For 1:

This part:

Since $F|(\mathbb{Z}/p\mathbb{Z})$ is a finite dimensional vector space, we must have $a_ku^k+\cdots+a_0=0,a_i\in \mathbb{Z}/p\mathbb{Z},1\leq i\leq k.$

Is a bit unclear to me. Where do the $a_i$ come from? What exactly are you using to conclude that $a_ku^k+\cdots+a_0=0$? And what is the number $k$?

For 2:

Suppose we have a "tower" $F\supseteq K \supseteq \mathbb{Z}/p\mathbb{Z}$. Then from part 1 we know that $F = (\mathbb{Z}/p\mathbb{Z})(u)$ for some $u\in F$. Now what can you say about $K(u)$ ?