/sci/ - Science & Math

I'm a bit confused at this, because I'm not too sure how the 5 is used, and I dont know if this is the only thing to it.

Let [math]\alpha[/math] be a root of [math]x^2-5[/math], and suppose it is not already in the base field [math]\mathbb F_p[/math]. Since this polynomial is over the field [math]\mathbb F_p[/math] already, we have that [math][\mathbb F_p(\alpha):\mathbb F_p]=2[/math]. Since there is only one field up to isomorphism of each particular order, then [math]F=F_p(\alpha)[/math]. QED

The 5 could have been just about any number, even 0, and it would still work with my alleged proof. Also the form of the polynomial was irrelevant. Following from that, with virtually the same proof, would generalize the statement to let [math]f(x)[/math] be any polynomial over [math]\mathbb F_p[/math]. Then it has a root in any field [math]F[/math] with [math]|F|=p^{\deg f}[/math].

Is this even remotely correct?