I just read about the following "fundamental" theorem of field extensions which is stated as follows:
Let $F$ be a field, and let $\alpha$ and $\beta$ be elements of field extensions $K/F$ and $L/F$. Suppose that $\alpha$ and $\beta$ are algebraic over $F$. There is an isomorphism of fields $\sigma: F(\alpha) \rightarrow F(\beta)$ that is the identity on $F$ and that sends $\alpha \mapsto \beta$ if and only if the irreducible polynomials for $\alpha$ and $\beta$ over $F$ are equal.
I was a little confused on what it means by what it means for two polynomials to be the same. My confusion stems from this example, where $F=\mathbb{F}_{3}$ and we consider $\mathbb{F}_{3}(\delta)$ where $\delta$ is the root of $x^{2}-2$. We have that $\eta= \delta +1$ is the root of $x^{2}+2x-1$. The field extensions $\mathbb{F}_{3}(\delta)$ and $\mathbb{F}_{3}(\eta)$ should be isomorphic by a map which sends $\delta$ to $\eta$. I don't quite see how these polynomials are the same. Did I miss something?
Yes, the part "the fields should be isomorphic" is not true. They are NOT isomporphic.
Assume that there is an isomorphism $F$ sending $\delta$ to $\eta$.
Then $$2=F(\delta^2)=F(\delta)^2=\eta^2$$
But this is not possible.