# Thread: Is any algebraically closed field infinite ?

1. ## Is any algebraically closed field infinite ?

Is the following statement true ?

If K is an algebraically closed field, then K is an infinite set.

2. We know that all finite fields are extensions of fields $\mathbb{Z}_p$. And in particular, any finite extension of them will not be algebraically closed. So the algebraic closure of $\mathbb{Z}_p$ is necessarily infinite.

3. Originally Posted by roninpro
We know that all finite fields are extensions of fields $\mathbb{Z}_p$. And in particular, any finite extension of them will not be algebraically closed. So the algebraic closure of $\mathbb{Z}_p$ is necessarily infinite.
Every finite field has order $p^n$ for some n. The underlying multiplicative group is always cyclic (of order $p^{n}-1$). Thus, $g^{p^n-1}=1$ for all $g \in \mathbb{F}_{n}\setminus \{0\}$.

Therefore, if $p \neq 2$, the equation $x^{p^n-1}+2$ will not have a root in $\mathbb{F}_n$.

If $p=2$ then this trick doesn't work...however, if $F=\{a_1, a_2, \ldots, a_{m}\}$ is your finite field, then the polynomial $(x-a_1)(x-a_2)\ldots (x-a_{m})+1$ has no roots in $F$ (this works for all p).

### algebraically closed field is infinite

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