Show that any finite extension of $\mathbb{Q}$ is not algebraically closed.

Question

How do I show that any finite extension of $\mathbb{Q}$ is not algebraically closed? In other words, the algebraic closure of $\mathbb{Q}$ is not a finite extension.

Answer 1

Let $\overline {\mathbb{Q}}_{\mathbb{C}}$ be the algebraic closure of $\mathbb{Q}$. Suppose the extension $\overline {\mathbb{Q}}_{\mathbb{C}}|\mathbb{Q}$ is finite. Then $[\overline {\mathbb{Q}}_{\mathbb{C}}:\mathbb{Q}]=n$ for certain $n\in\mathbb{N}$. If $m:=n+1$ and $a:=2^{1/m}$, $a\in\overline {\mathbb{Q}}_{\mathbb{C}}$ because $a$ is a root of the polynomial $f(t):=t^{m}-2$. This polynomial is irreducible in $\mathbb{Z}[t]$ by Eisenstein's criterion, so it is also irreducible in $\mathbb{Q}[t]$ since $f\in\mathbb{Z}[t]$ and $\mathbb{Q}$ is the quotient field of $\mathbb{Z}$. According to this, $f$ is the minimal polynomial of $a$ in $\mathbb{Q}[t]$ and there is a contradiction because: $m=deg(f)=[\mathbb{Q}(a):\mathbb{Q}]\leq[\overline {\mathbb{Q}}_{\mathbb{C}}:\mathbb{Q}]=n=m-1$

Answer 2

Edit: This concerns the old question. We have the following result:

Theorem: If $K$ is a countable field, then its algebraic closure is countable. In particular the algebraic closures of the prime fields $\mathbb{Q}$ and $\mathbb{F}_p$ are countable.

For the new question: The degree $[\overline{\mathbb{Q}}:\mathbb{Q}]$ cannot be finite, because we have irreducible polynomials $x^n-a$ of arbitrary high degree. That the algebraic closure is not finite, as you have written, is clear anyway, because it contains an infinite field, i.e., $\mathbb{Q}$.

Answer 3

You can for example show that the algebraic closure of $\mathbb{Q}$ contains arbitrarily large finite $\mathbb{Q}$-linearly independent set.

Pick your favourite one, mine would be the sets $1, \sqrt[n]{2}, \left( \sqrt[n]{2} \right)^2, \dots, \left( \sqrt[n]{2} \right)^{n-1}$ for all choices of $n$. Then the irreducibility of the polynomial $x^n-2$ (by Eisenstein criterion) can be used to see the linear independence.

(Apologies for the edit, this is the example I actually had in mind.)