Let $F$ be a Field with $\mathrm{char}(F) > 0$. $E_1$ and $E_2$ are finite extensions of $F$ of same degree. Prove that $E_1$ and $E_2$ may not be isomorphic.
In Fields of characteristic zero, I get the obvious counter example namely $\mathbf{Q}(\sqrt2)$ and $\mathbf{Q}(\sqrt3)$. In finite (Perfect) Fields, the result is true. But I am unable to get a counter example for an infinite Field of prime characteristic. I need help from someone please!!
Finite Field extensions of same degree need not be isomorphic as Fields.
abstract-algebra
Best Answer
Edit: sorry misread the question! For instance take $\mathbf{F}_{p^2}((t))/\mathbf{F}_p((t))$ and $\mathbf{F}_p((t^{1/2}))/\mathbf{F}_p((t))$. These are both degree 2 extensions, but are not isomorphic: in particular the second one is isomorphic to $\mathbf{F}_p((t))$ itself, which is not isomorphic to $\mathbf{F}_{p^2}((t))$.
Let's show that these are degree 2 extensions. Note $\mathbf{F}_{p^2}/\mathbf{F}_p$ is a degree $2$ extension, given by adjoining a root of an irreducible polynomial $f(x) \in \mathbf{F}_p[x]$ of degree $2$ over $\mathbf{F}_p$ (for instance, if $p$ is odd, take $x^2 - r$, where $r \in \mathbf{F}_p$ is not a quadratic residue mod $p$). Then the extension $\mathbf{F}_{p^2}((t))/\mathbf{F}_p((t))$ is given by, again, adjoining a root of $f(x)$, this time viewed as a polynomial in $\mathbf{F}_p((t))[x]$: note it's still irreducible in this ring, and thus the extension is of degree $2$. On the other hand, you could adjoin a root of the polynomial $x^2 - t \in \mathbf{F}_p((t))[x]$ and then you get the extension $\mathbf{F}_p((t^{1/2}))$: this is again of degree $2$ since the polynomial has degree $2$.
More generally this illustrates the distinction between ramified and unramified extensions of equal characteristic local fields.
You could also do this starting with $\mathbf{F}_p(t)$, the field of rational functions in one variable over $\mathbf{F}_p$, instead of $\mathbf{F}_p((t))$ the field of Laurent series in one variable over $\mathbf{F}_p$, it wouldn't change anything above, and you could come up with more examples. I just chose Laurent series because the theory of finite field extensions of a local fields is a bit simpler in the end.