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This is really just a yes/no question, but I need to make sure that I am write about the following.

If $\alpha$ is algebraic over $F$, then $F[\alpha]$ is a field. To show this, I let $f(x)$ be the minimal polynomial for $F$ for $\alpha$ over $F$. Then the ideal generated by $f$ is maximal. Then one can show that there is an isomorphism $$ F[x] / \langle f(x)\rangle $$ and $$ F[\alpha]. $$ And this shows that $F[\alpha]$ is a field.

Is this a correct way to do this?

How might I prove that $F[\beta]$ is a field only if $\beta$ is algebraic over $F$?

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Yes you are correct that that is a way to show $F[\alpha]$ is a field. To show that $F[\beta]$ is a field implies $\beta$ is algebraic, note that $\frac{1}{\beta}$ must be in $F[\beta]$, and thus is a polynomial in $\beta$. What can you do from here?

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If it's a field you must have an equation of the form $$\frac{1}{a}=c_0+\cdots +c_na^n$$

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