Can someone explain the following excerpt to me:
In fact, all $\Bbb F_{p^r}$ together form the algebraic closure of $\Bbb F_p$.
and tell me what $\Bbb Q_p$ is? I am aware of the $p$-adic integers $\Bbb Z_p$, but not of $\Bbb Q_p$.
The first was said, and the latter was written in a talk I went to months ago, and I was just looking through the notes I took.
On
In the algebraic closure $\overline{\mathbb F_p}$ of $\mathbb F_p$, you have one subfield isomorphic to $\mathbb F_{p^k}$ for each $k$, namely:
$$A_k=\{x\in \overline{\mathbb F_p}\mid x^{p^k}=x\}$$
Since there are no repeated roots to $x^{p^k}-x$ (why?) and the polynomial splits in $\overline{\mathbb F_p}$, you must have $p^k$ distinct roots. Show that they form a field, and hence $A_k\cong\mathbb F_{p^k}.$
Now, ever element of $\overline{\mathbb F_p}$ is the root of some irreducible polynomial of degree, say, $k$, but then that element must be in $A_k\cong \mathbb F_{p^k}.$ So every element of $\overline{\mathbb F_p}$ is in some $A_k.$ So $\overline{\mathbb F_p}=\bigcup_{k=1}^{\infty} A_k.$
$\mathbb Q_p$ are the $p$-adic numbers. They can be considered as any of:
$\mathbb{Q}_p$ is the field of $p$-adic numbers, which is the field of fractions of $\mathbb{Z}_p$ (or if you prefer, the completion of $\mathbb{Q}$ with respect to the $p$-adic absolute value). The highlighted statement is just the fact that $$ \overline{\mathbb{F}}_p=\bigcup_{r\geq 1}\mathbb{F}_{p^r}$$